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:

$$\frac{1}{x^{5}} = 0$$

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:

$$\frac{1}{x^{5}} = 0$$

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

$$- \frac{5}{x^{6}} = 0$$

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

$$- \frac{5}{x^{6}} = 0$$

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

$$\frac{30}{x^{7}} = 0$$

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

$$\frac{30}{x^{7}} = 0$$

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

$$\lim_{x \to -\infty} \frac{1}{x^{5}} = 0$$

Let's take the limit

so,

equation of the horizontal asymptote on the left:

$$y = 0$$

$$\lim_{x \to \infty} \frac{1}{x^{5}} = 0$$

Let's take the limit

so,

equation of the horizontal asymptote on the right:

$$y = 0$$

$$\lim_{x \to -\infty} \frac{1}{x^{5}} = 0$$

Let's take the limit

so,

equation of the horizontal asymptote on the left:

$$y = 0$$

$$\lim_{x \to \infty} \frac{1}{x^{5}} = 0$$

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 x^(-5), divided by x at x->+oo and x ->-oo

$$\lim_{x \to -\infty} \frac{1}{x^{6}} = 0$$

Let's take the limit

so,

inclined coincides with the horizontal asymptote on the right

$$\lim_{x \to \infty} \frac{1}{x^{6}} = 0$$

Let's take the limit

so,

inclined coincides with the horizontal asymptote on the left

$$\lim_{x \to -\infty} \frac{1}{x^{6}} = 0$$

Let's take the limit

so,

inclined coincides with the horizontal asymptote on the right

$$\lim_{x \to \infty} \frac{1}{x^{6}} = 0$$

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:

$$\frac{1}{x^{5}} = - \frac{1}{x^{5}}$$

- No

$$\frac{1}{x^{5}} = \frac{1}{x^{5}}$$

- Yes

so, the function

is

odd

So, check:

$$\frac{1}{x^{5}} = - \frac{1}{x^{5}}$$

- No

$$\frac{1}{x^{5}} = \frac{1}{x^{5}}$$

- Yes

so, the function

is

odd