Question

Determine intervals in which following functions are strictly increasing or strictly decreasing.
$f:R\to R,f\left(x\right)=x^3-6x^2-36x+2$.

Answer 1

$f:R\to R$, $f\left(x\right)=x^3-6x^2-36x+2$

$f^{\prime }\left(x\right)=3x^2-12x-36$

$=3(x^2-4x-12)$

$=3(x^2-6x+2x-12)$

$=3(x+2)(x-6)$

$f\left(x\right)$ is strictly increasing if $f^{\prime }\left(x\right)>0$

$3(x+2)(x-6)>0$

$x<-2$ and $x>6$

$f\left(x\right)$ is strictly decreasing if $f^{\prime }\left(x\right)<0$

$3(x+2)(x-6)<0$

$-2<x<6$