# How do you use the first derivative to determine where the function f(x)= 3 x^4 + 96 x is increasing or decreasing?

Nov 25, 2016

$f \left(x\right)$ is decreasing when x in ] -oo,-2 ]
$f \left(x\right)$ is increasing when x in[-2, +oo[

#### Explanation:

$f \left(x\right) = 3 {x}^{4} + 96 x$

The derivative is,
$f ' \left(x\right) = 12 {x}^{3} + 96$

$f ' \left(x\right) = 0$

when, $12 {x}^{3} + 96 = 0$

$12 {x}^{3} = - 96$

${x}^{3} = - 8$

$x = - 2$

Let's do a sign chart,

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

$\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}$$-$$\textcolor{w h i t e}{a a a a a}$$0$$\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}$$\downarrow$$\textcolor{w h i t e}{a a a}$$- 144$$\textcolor{w h i t e}{a a a a}$$\uparrow$

So, $f \left(x\right)$ is decreasing when x in ] -oo,-2 ]

and $f \left(x\right)$ is increasing when x in[-2, +oo[