Answer to Question #326690 in Calculus for Vava

Question

study the function : x^2/x-1

Answer 1

In order to find the extrema, you need to solve the equation

$\frac{d}{d x} f{\left(x \right)} = 0$

(the derivative is zero),

and the roots of this equation will be the extrema of this function:

$\frac{d}{d x} f{\left(x \right)} =$

first derivative

$- \frac{x^{2}}{\left(x - 1\right)^{2}} + \frac{2 x}{x - 1} = 0$

We solve this equation

The roots of this equation

$x_{1} = 0$

$x_{2} = 2$

Zn. extremes at points:

(0, 0)

(2, 4)

Intervals of increasing and decreasing functions:

Let's find the intervals where the function increases and decreases, as well as the minima and maxima of the function, for this we look at how the function behaves at extrema with the slightest deviation from the extremum:

Minima of the function at points:

$x_{1} = 2$

Function maxima at points:

$x_{1} = 0$

Falling off in between

$\left(-\infty, 0\right] \cup \left[2, \infty\right)$

Increasing in between

$\left[0, 2\right]$

We find horizontal asymptotes using the limits of this function for x->+oo and x->-oo

$\lim_{x \to -\infty}\left(\frac{x^{2}}{x - 1}\right) = -\infty$

Take the limit

, which means that

there is no horizontal asymptote on the left

$\lim_{x \to \infty}\left(\frac{x^{2}}{x - 1}\right) = \infty$

Take the limit

, which means that

there is no horizontal asymptote on the right

We find the inflection points, for this we need to solve the equation

$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$

(the second derivative is zero),

the roots of the resulting equation will be the inflection points for the specified function graph:

$\frac{d^{2}}{d x^{2}} f{\left(x \right)} =$

second derivative

$\frac{2 \left(\frac{x^{2}}{\left(x - 1\right)^{2}} - \frac{2 x}{x - 1} + 1\right)}{x - 1} = 0$

We solve this equation

No solutions were found,

the function may not have inflections

The oblique asymptote can be found by calculating the limit of the function x^2/(x - 1) divided by x at x->+oo and x ->-oo

$\lim_{x \to -\infty}\left(\frac{x}{x - 1}\right) = 1$

Let's take the limit

, which means

the equation of the oblique asymptote on the left:

Y=x

$\lim_{x \to \infty}\left(\frac{x}{x - 1}\right) = 1$

Let's take the limit

, which means

the equation of the oblique asymptote on the right:

Y=x

Let's check if the function is even or odd using the relations f = f(-x) and f = -f(-x).

So let's check:

$\frac{x^{2}}{x - 1} = \frac{x^{2}}{- x - 1}$

- Not

$\frac{x^{2}}{x - 1} = - \frac{x^{2}}{- x - 1}$

- No

means the function

is

neither even nor odd

Points at which a function is exactly undefined: $x_{1} = 1$

Vertical asymptotes

There is: $x_{1} = 1$

The graph of the function intersects the X axis at f = 0,

so you need to solve the equation:

$\frac{x^{2}}{x - 1} = 0$

Solving this equation

Intersections with the X-axis:

Analytical solution

$x_{1} = 0$

Numerical solution

$x_{1} = 0$

The graph crosses the y-axis when x equals 0:

substitute x = 0 into x²/(x - 1)

$\frac{0^{2}}{-1}$

Result:

$f{\left(0 \right)} = 0$

Dot:

(0, 0)