Mister Exam
Graphing y = log10(x)-arctg(x)
The solution
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log(x)
f(x) = ------- - atan(x)
log(10)
$$f{\left(x \right)} = \frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}$$
f = log(x)/log(10) - atan(x)
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 34.8420115290269$$
so we need to solve the equation:
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 34.8420115290269$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{1}{x^{2} + 1} + \frac{1}{x \log{\left(10 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
$$x_{2} = \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
The values of the extrema at the points:
Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Maxima of the function at points:
$$x_{1} = - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Decreasing at intervals
$$\left(-\infty, - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}\right] \cup \left[\frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}, \infty\right)$$
Increasing at intervals
$$\left[- \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}, \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{1}{x^{2} + 1} + \frac{1}{x \log{\left(10 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
$$x_{2} = \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
The values of the extrema at the points:
/ _______________\
| / 2 |
_______________ |log(10) \/ -4 + log (10) | / _______________ \
/ 2 log|------- - ------------------| | / 2 |
log(10) \/ -4 + log (10) \ 2 2 / |\/ -4 + log (10) log(10)|
(------- - ------------------, --------------------------------- + atan|------------------ - -------|)
2 2 log(10) \ 2 2 / / _______________ \
| / 2 |
_______________ / _______________ \ |\/ -4 + log (10) log(10)|
/ 2 | / 2 | log|------------------ + -------|
\/ -4 + log (10) log(10) |\/ -4 + log (10) log(10)| \ 2 2 /
(------------------ + -------, - atan|------------------ + -------| + ---------------------------------)
2 2 \ 2 2 / log(10) Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Maxima of the function at points:
$$x_{1} = - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Decreasing at intervals
$$\left(-\infty, - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}\right] \cup \left[\frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}, \infty\right)$$
Increasing at intervals
$$\left[- \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}, \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}\right]$$
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{2 x}{\left(x^{2} + 1\right)^{2}} - \frac{1}{x^{2} \log{\left(10 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}$$
$$x_{2} = \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[- \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}, \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}\right]$$
Convex at the intervals
$$\left(-\infty, - \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}\right] \cup \left[\frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{2 x}{\left(x^{2} + 1\right)^{2}} - \frac{1}{x^{2} \log{\left(10 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}$$
$$x_{2} = \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[- \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}, \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}\right]$$
Convex at the intervals
$$\left(-\infty, - \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}\right] \cup \left[\frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(x)/log(10) - atan(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = \frac{\log{\left(- x \right)}}{\log{\left(10 \right)}} + \operatorname{atan}{\left(x \right)}$$
- No
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = - \frac{\log{\left(- x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = \frac{\log{\left(- x \right)}}{\log{\left(10 \right)}} + \operatorname{atan}{\left(x \right)}$$
- No
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = - \frac{\log{\left(- x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd