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

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Answer 1 · 2016-11-25T09:11:30

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

$f(x)=3x^4+96x$

The derivative is,
$f'(x)=12x^3+96$

$f'(x)=0$

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

$12x^3=-96$

$x^3=-8$

$x=-2$

Let's do a sign chart,

$color(white)(aaaa)$ $x$ $color(white)(aaaaa)$ $-oo$ $color(white)(aaaaaa)$ $-2$ $color(white)(aaaaa)$ $+oo$

$color(white)(aaaa)$ $f'(x)$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaaa)$ $0$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ $darr$ $color(white)(aaa)$ $-144$ $color(white)(aaaa)$ $uarr$

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

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

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