Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=2x^3-24x+107$

Answer 1

When $(x-a)(x-b)>0$ with $a<b,$ $x<a$ or $x>b.$

When $(x-a)(x-b)<0$ with $a<b,a<x<b.$

$f\left(x\right)=2x^3-24x+107$ [ Given ]

Differentiating both sides,

$\Rightarrow$ $f^{\prime }\left(x\right)=6x^2-24$

$\Rightarrow$ $f^{\prime }\left(x\right)=6(x^2-4)$

$\Rightarrow$ $f^{\prime }\left(x\right)=6(x+2)(x-2)$

For $f\left(x\right)$ to be increasing, we must have

$f^{\prime }\left(x\right)>0$

$\Rightarrow$ $6(x+2)(x-2)>0$

$\Rightarrow$ $(x+2)(x-2)>0$

$\Rightarrow$ $x<-2$ or $x>2$

$\Rightarrow$ $x\in (-\infty ,-2)\cup (2,\infty )$

So, $f\left(x\right)$ is increasing on $x\in (-\infty ,-2)\cup (2,\infty ).$

For $f\left(x\right)$ to be decreasing, we must have

$f^{\prime }\left(x\right)<0$

$\Rightarrow$ $6(x+2)(x-2)<0$

$\Rightarrow$ $(x+2)(x-2)<0$

$\Rightarrow$ $-2<x<2$

$\Rightarrow$ $x\in (-2,2)$

So, $f\left(x\right)$ is decreasing on $x\in (-2,2).$