Mister Exam
Graphing y = x*exp(2*x)*log(x)
The solution
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2*x f(x) = x*e *log(x)
$$f{\left(x \right)} = x e^{2 x} \log{\left(x \right)}$$
f = (x*exp(2*x))*log(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:
$$x e^{2 x} \log{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$x e^{2 x} \log{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$\left(2 x e^{2 x} + e^{2 x}\right) \log{\left(x \right)} + e^{2 x} = 0$$
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
$$\left(2 x e^{2 x} + e^{2 x}\right) \log{\left(x \right)} + e^{2 x} = 0$$
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
$$\left(4 \left(x + 1\right) \log{\left(x \right)} + \frac{2 \left(2 x + 1\right)}{x} - \frac{1}{x}\right) e^{2 x} = 0$$
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
$$\left(4 \left(x + 1\right) \log{\left(x \right)} + \frac{2 \left(2 x + 1\right)}{x} - \frac{1}{x}\right) e^{2 x} = 0$$
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
$$\lim_{x \to -\infty}\left(x e^{2 x} \log{\left(x \right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(x e^{2 x} \log{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x e^{2 x} \log{\left(x \right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(x e^{2 x} \log{\left(x \right)}\right) = \infty$$
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 (x*exp(2*x))*log(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(e^{2 x} \log{\left(x \right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(e^{2 x} \log{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(e^{2 x} \log{\left(x \right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(e^{2 x} \log{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$x e^{2 x} \log{\left(x \right)} = - x e^{- 2 x} \log{\left(- x \right)}$$
- No
$$x e^{2 x} \log{\left(x \right)} = x e^{- 2 x} \log{\left(- x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x e^{2 x} \log{\left(x \right)} = - x e^{- 2 x} \log{\left(- x \right)}$$
- No
$$x e^{2 x} \log{\left(x \right)} = x e^{- 2 x} \log{\left(- x \right)}$$
- No
so, the function
not is
neither even, nor odd