Mister Exam
Graphing y = log(3*x)+sqrt(x)
The solution
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___ f(x) = log(3*x) + \/ x
$$f{\left(x \right)} = \sqrt{x} + \log{\left(3 x \right)}$$
f = sqrt(x) + log(3*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:
$$\sqrt{x} + \log{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{3 e^{2 W\left(\frac{\sqrt{3}}{6}\right)}}$$
Numerical solution
$$x_{1} = 0.210646363022657$$
so we need to solve the equation:
$$\sqrt{x} + \log{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{3 e^{2 W\left(\frac{\sqrt{3}}{6}\right)}}$$
Numerical solution
$$x_{1} = 0.210646363022657$$
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
$$\frac{1}{x} + \frac{1}{2 \sqrt{x}} = 0$$
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
$$\frac{1}{x} + \frac{1}{2 \sqrt{x}} = 0$$
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
$$- (\frac{1}{x^{2}} + \frac{1}{4 x^{\frac{3}{2}}}) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- (\frac{1}{x^{2}} + \frac{1}{4 x^{\frac{3}{2}}}) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sqrt{x} + \log{\left(3 x \right)}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\sqrt{x} + \log{\left(3 x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\sqrt{x} + \log{\left(3 x \right)}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\sqrt{x} + \log{\left(3 x \right)}\right) = \infty$$
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(3*x) + sqrt(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x} + \log{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{x} + \log{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x} + \log{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{x} + \log{\left(3 x \right)}}{x}\right) = 0$$
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:
$$\sqrt{x} + \log{\left(3 x \right)} = \sqrt{- x} + \log{\left(- 3 x \right)}$$
- No
$$\sqrt{x} + \log{\left(3 x \right)} = - \sqrt{- x} - \log{\left(- 3 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{x} + \log{\left(3 x \right)} = \sqrt{- x} + \log{\left(- 3 x \right)}$$
- No
$$\sqrt{x} + \log{\left(3 x \right)} = - \sqrt{- x} - \log{\left(- 3 x \right)}$$
- No
so, the function
not is
neither even, nor odd