Mister Exam
Graphing y = (2*ln(x+3)/x)-3
The solution
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2*log(x + 3)
f(x) = ------------ - 3
x
$$f{\left(x \right)} = -3 + \frac{2 \log{\left(x + 3 \right)}}{x}$$
f = -3 + (2*log(x + 3))/x
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3 - \frac{2 W\left(- \frac{3}{2 e^{\frac{9}{2}}}\right)}{3}$$
$$x_{2} = -3 - \frac{2 W_{-1}\left(- \frac{3}{2 e^{\frac{9}{2}}}\right)}{3}$$
Numerical solution
$$x_{1} = -2.98870112007792$$
$$x_{2} = 0.908824451300817$$
so we need to solve the equation:
$$-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3 - \frac{2 W\left(- \frac{3}{2 e^{\frac{9}{2}}}\right)}{3}$$
$$x_{2} = -3 - \frac{2 W_{-1}\left(- \frac{3}{2 e^{\frac{9}{2}}}\right)}{3}$$
Numerical solution
$$x_{1} = -2.98870112007792$$
$$x_{2} = 0.908824451300817$$
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
$$\frac{2}{x \left(x + 3\right)} - \frac{2 \log{\left(x + 3 \right)}}{x^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{2}{x \left(x + 3\right)} - \frac{2 \log{\left(x + 3 \right)}}{x^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
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
$$\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 54232.7720154785$$
$$x_{2} = 47557.9021762819$$
$$x_{3} = 55341.460585528$$
$$x_{4} = 40837.733275076$$
$$x_{5} = 46441.2859799058$$
$$x_{6} = 48673.2683225189$$
$$x_{7} = 49787.4275369804$$
$$x_{8} = 50900.4203746904$$
$$x_{9} = 41961.3545187714$$
$$x_{10} = 57555.8805564298$$
$$x_{11} = 35194.474176351$$
$$x_{12} = 39712.5339023989$$
$$x_{13} = 32923.8658113025$$
$$x_{14} = 30644.4070335614$$
$$x_{15} = 58661.6683613746$$
$$x_{16} = -2$$
$$x_{17} = 52012.285038714$$
$$x_{18} = 44204.1166985693$$
$$x_{19} = 56449.1537881797$$
$$x_{20} = 53123.0575695818$$
$$x_{21} = 37457.1165432152$$
$$x_{22} = 36326.741841763$$
$$x_{23} = 38585.6867389593$$
$$x_{24} = 45323.3738151654$$
$$x_{25} = 43083.4622826575$$
$$x_{26} = 34060.2172304577$$
$$x_{27} = 31785.3046573135$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
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
$$\left(-\infty, -2\right]$$
Convex at the intervals
$$\left[-2, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 54232.7720154785$$
$$x_{2} = 47557.9021762819$$
$$x_{3} = 55341.460585528$$
$$x_{4} = 40837.733275076$$
$$x_{5} = 46441.2859799058$$
$$x_{6} = 48673.2683225189$$
$$x_{7} = 49787.4275369804$$
$$x_{8} = 50900.4203746904$$
$$x_{9} = 41961.3545187714$$
$$x_{10} = 57555.8805564298$$
$$x_{11} = 35194.474176351$$
$$x_{12} = 39712.5339023989$$
$$x_{13} = 32923.8658113025$$
$$x_{14} = 30644.4070335614$$
$$x_{15} = 58661.6683613746$$
$$x_{16} = -2$$
$$x_{17} = 52012.285038714$$
$$x_{18} = 44204.1166985693$$
$$x_{19} = 56449.1537881797$$
$$x_{20} = 53123.0575695818$$
$$x_{21} = 37457.1165432152$$
$$x_{22} = 36326.741841763$$
$$x_{23} = 38585.6867389593$$
$$x_{24} = 45323.3738151654$$
$$x_{25} = 43083.4622826575$$
$$x_{26} = 34060.2172304577$$
$$x_{27} = 31785.3046573135$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
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
$$\left(-\infty, -2\right]$$
Convex at the intervals
$$\left[-2, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(-3 + \frac{2 \log{\left(x + 3 \right)}}{x}\right) = -3$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -3$$
$$\lim_{x \to \infty}\left(-3 + \frac{2 \log{\left(x + 3 \right)}}{x}\right) = -3$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -3$$
$$\lim_{x \to -\infty}\left(-3 + \frac{2 \log{\left(x + 3 \right)}}{x}\right) = -3$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -3$$
$$\lim_{x \to \infty}\left(-3 + \frac{2 \log{\left(x + 3 \right)}}{x}\right) = -3$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -3$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (2*log(x + 3))/x - 3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-3 + \frac{2 \log{\left(x + 3 \right)}}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-3 + \frac{2 \log{\left(x + 3 \right)}}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{-3 + \frac{2 \log{\left(x + 3 \right)}}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-3 + \frac{2 \log{\left(x + 3 \right)}}{x}}{x}\right) = 0$$
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:
$$-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = -3 - \frac{2 \log{\left(3 - x \right)}}{x}$$
- No
$$-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = 3 + \frac{2 \log{\left(3 - x \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = -3 - \frac{2 \log{\left(3 - x \right)}}{x}$$
- No
$$-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = 3 + \frac{2 \log{\left(3 - x \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd