Mister Exam
Graphing y = exp*ln(x)/(sqrt(x))
The solution
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x
e *log(x)
f(x) = ---------
___
\/ x
$$f{\left(x \right)} = \frac{e^{x} \log{\left(x \right)}}{\sqrt{x}}$$
f = (exp(x)*log(x))/sqrt(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:
$$\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}} = 0$$
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:
$$\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}} = 0$$
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
$$\frac{e^{x} \log{\left(x \right)} + \frac{e^{x}}{x}}{\sqrt{x}} - \frac{e^{x} \log{\left(x \right)}}{2 x^{\frac{3}{2}}} = 0$$
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
$$\frac{e^{x} \log{\left(x \right)} + \frac{e^{x}}{x}}{\sqrt{x}} - \frac{e^{x} \log{\left(x \right)}}{2 x^{\frac{3}{2}}} = 0$$
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
$$\frac{\left(\log{\left(x \right)} - \frac{\log{\left(x \right)} + \frac{1}{x}}{x} + \frac{2}{x} + \frac{3 \log{\left(x \right)}}{4 x^{2}} - \frac{1}{x^{2}}\right) e^{x}}{\sqrt{x}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 1$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(\log{\left(x \right)} - \frac{\log{\left(x \right)} + \frac{1}{x}}{x} + \frac{2}{x} + \frac{3 \log{\left(x \right)}}{4 x^{2}} - \frac{1}{x^{2}}\right) e^{x}}{\sqrt{x}}\right) = \infty i$$
$$\lim_{x \to 0^+}\left(\frac{\left(\log{\left(x \right)} - \frac{\log{\left(x \right)} + \frac{1}{x}}{x} + \frac{2}{x} + \frac{3 \log{\left(x \right)}}{4 x^{2}} - \frac{1}{x^{2}}\right) e^{x}}{\sqrt{x}}\right) = -\infty$$
- 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
$$\left[1, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 1\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(\log{\left(x \right)} - \frac{\log{\left(x \right)} + \frac{1}{x}}{x} + \frac{2}{x} + \frac{3 \log{\left(x \right)}}{4 x^{2}} - \frac{1}{x^{2}}\right) e^{x}}{\sqrt{x}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 1$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(\log{\left(x \right)} - \frac{\log{\left(x \right)} + \frac{1}{x}}{x} + \frac{2}{x} + \frac{3 \log{\left(x \right)}}{4 x^{2}} - \frac{1}{x^{2}}\right) e^{x}}{\sqrt{x}}\right) = \infty i$$
$$\lim_{x \to 0^+}\left(\frac{\left(\log{\left(x \right)} - \frac{\log{\left(x \right)} + \frac{1}{x}}{x} + \frac{2}{x} + \frac{3 \log{\left(x \right)}}{4 x^{2}} - \frac{1}{x^{2}}\right) e^{x}}{\sqrt{x}}\right) = -\infty$$
- 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
$$\left[1, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 1\right]$$
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
$$\lim_{x \to -\infty}\left(\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}}\right) = \infty$$
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 (exp(x)*log(x))/sqrt(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{x} \log{\left(x \right)}}{\sqrt{x} x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{x} \log{\left(x \right)}}{\sqrt{x} x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{e^{x} \log{\left(x \right)}}{\sqrt{x} x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{x} \log{\left(x \right)}}{\sqrt{x} x}\right) = \infty$$
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:
$$\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}} = \frac{e^{- x} \log{\left(- x \right)}}{\sqrt{- x}}$$
- No
$$\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}} = - \frac{e^{- x} \log{\left(- x \right)}}{\sqrt{- x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}} = \frac{e^{- x} \log{\left(- x \right)}}{\sqrt{- x}}$$
- No
$$\frac{e^{x} \log{\left(x \right)}}{\sqrt{x}} = - \frac{e^{- x} \log{\left(- x \right)}}{\sqrt{- x}}$$
- No
so, the function
not is
neither even, nor odd