Mister Exam
Graphing y = exp(-sqrt(x))/(sqrt(x))
The solution
You have entered
[src]
___
-\/ x
e
f(x) = -------
___
\/ x
$$f{\left(x \right)} = \frac{e^{- \sqrt{x}}}{\sqrt{x}}$$
f = exp(-sqrt(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:
$$\frac{e^{- \sqrt{x}}}{\sqrt{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{e^{- \sqrt{x}}}{\sqrt{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$- \frac{e^{- \sqrt{x}}}{2 \sqrt{x} \sqrt{x}} - \frac{e^{- \sqrt{x}}}{2 x^{\frac{3}{2}}} = 0$$
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
$$- \frac{e^{- \sqrt{x}}}{2 \sqrt{x} \sqrt{x}} - \frac{e^{- \sqrt{x}}}{2 x^{\frac{3}{2}}} = 0$$
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
$$\frac{\left(\frac{2}{x^{2}} + \frac{\frac{1}{x} + \frac{1}{x^{\frac{3}{2}}}}{\sqrt{x}} + \frac{3}{x^{\frac{5}{2}}}\right) e^{- \sqrt{x}}}{4} = 0$$
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
$$\frac{\left(\frac{2}{x^{2}} + \frac{\frac{1}{x} + \frac{1}{x^{\frac{3}{2}}}}{\sqrt{x}} + \frac{3}{x^{\frac{5}{2}}}\right) e^{- \sqrt{x}}}{4} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x}}\right)$$
$$\lim_{x \to \infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x}}\right)$$
$$\lim_{x \to \infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of exp(-sqrt(x))/sqrt(x), divided by x at x->+oo and x ->-oo
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x} x}\right)$$
$$\lim_{x \to \infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x} x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x} x}\right)$$
$$\lim_{x \to \infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x} x}\right) = 0$$
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:
$$\frac{e^{- \sqrt{x}}}{\sqrt{x}} = \frac{e^{- \sqrt{- x}}}{\sqrt{- x}}$$
- No
$$\frac{e^{- \sqrt{x}}}{\sqrt{x}} = - \frac{e^{- \sqrt{- x}}}{\sqrt{- x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{e^{- \sqrt{x}}}{\sqrt{x}} = \frac{e^{- \sqrt{- x}}}{\sqrt{- x}}$$
- No
$$\frac{e^{- \sqrt{x}}}{\sqrt{x}} = - \frac{e^{- \sqrt{- x}}}{\sqrt{- x}}$$
- No
so, the function
not is
neither even, nor odd