Answer to Question #325171 in Calculus for Math

Question

Let f(x)= x^2/x^2-9

(a) Find the intervals where f is increasing and decreasing. Identify the relative

extrema.

(b) Find the intervals where f is concave up and down and identify any inflection points.

(c) Sketch a graph of f using the information from this problem

Answer 1

a. 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

$- \frac{2 x^{3}}{\left(x^{2} - 9\right)^{2}} + \frac{2 x}{x^{2} - 9} =0$

We solve this equation

The roots of this equation

$x_{1} = 0$

Zn. extremes at points:

(0, 0)

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: The

function has no minima Maximums of the

function at points:

$x_{1} = 0$

Falling off in between$\left(-\infty, 0\right]$

Increasing in between$\left[0, \infty\right)$

b. 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

$\frac{2 \left(\frac{x^{2} \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{4 x^{2}}{x^{2} - 9} + 1\right)}{x^{2} - 9} = 0$

We solve this equation

No solutions were found,

the function may not have inflections

c.