# How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for f(x)=(x^3)/(x^2-4)?

Apr 14, 2017

The intervals of increasing are $x \in \left(- \infty , - \sqrt{12}\right) \cup \left(\sqrt{12} , + \infty\right)$
The intervals of decreasing are $x \in \left(- \sqrt{12} , - 2\right) \cup \left(- 2 , + 2\right) \cup \left(2 , \sqrt{12}\right)$

#### Explanation:

We calculate the first derivative and construct a sign chart.

We need

$\left(\frac{u}{v}\right) ' = \frac{u ' v - u v '}{{v}^{2}}$

$f \left(x\right) = {x}^{3} / \left({x}^{2} - 4\right)$

$u = {x}^{3}$, $\implies$, $u ' = 3 {x}^{2}$

$v = {x}^{2} - 4$, $\implies$, $v ' = 2 x$

$f ' \left(x\right) = \frac{3 {x}^{2} \left({x}^{2} - 4\right) - 2 x \cdot {x}^{3}}{{x}^{2} - 4} ^ 2$

$= \frac{3 {x}^{4} - 12 {x}^{2} - 2 {x}^{4}}{{x}^{2} - 4} ^ 2$

$= \frac{{x}^{4} - 12 {x}^{2}}{{x}^{2} - 4} ^ 2$

$= \frac{{x}^{2} \left({x}^{2} - 12\right)}{{x}^{2} - 4} ^ 2$

$f ' \left(x\right) = \frac{{x}^{2} \left(x + \sqrt{12}\right) \left(x - \sqrt{12}\right)}{{\left(x + 2\right)}^{2} {\left(x - 2\right)}^{2}}$

We construct the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- \sqrt{12}$$\textcolor{w h i t e}{a a a}$$- 2$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$2$$\textcolor{w h i t e}{a a a}$$\sqrt{12}$$\textcolor{w h i t e}{a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + \sqrt{12}$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a}$$+$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x + 2$$\textcolor{w h i t e}{a a a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a}$$+$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 2$$\textcolor{w h i t e}{a a a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a}$$+$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - \sqrt{12}$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a}$$-$$\textcolor{w h i t e}{a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f ' \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a}$$-$$\textcolor{w h i t e}{a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a a}$↗$\textcolor{w h i t e}{a a a a}$↘$\textcolor{w h i t e}{a a a}$↘$\textcolor{w h i t e}{a}$↘$\textcolor{w h i t e}{a a}$↘$\textcolor{w h i t e}{a a a a}$↗

The relative maximum is when $x = - \sqrt{12}$

The relative minimum is when $x = \sqrt{12}$

graph{x^3/(x^2-4) [-14.24, 14.24, -7.12, 7.12]}