Question

Determine intervals in which following functions are strictly increasing or strictly decreasing:
$\left(1\right)$ $f:R\to R,f\left(x\right)=x^3-6x^2-36x+2$
$\left(2\right)$ $f:R\to R,f\left(x\right)=x^4-4x$.

Answer 1

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

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

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

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

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

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

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

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

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

$\Rightarrow$ $-2<x<6$

(2) $f:R\to R$, $f\left(x\right)=x^4-4x$

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

$=4(x-1)(x^2+x+1)$

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

$\Rightarrow$ $x>1$

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

$\Rightarrow$ $x<1$