Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=-2x^3-9x^2-12x+1$

Answer 1

We have

$f\left(x\right)=-2x^3-9x^2-12x+1$

$f^{\prime }\left(x\right)=-6x^2-18x-12$

For critical points, put $f^{\prime }\left(x\right)=0$

i.e., $-6x^2-18x-12=0$

$-6(x^2+3x+2)=0$

$(x+2)(x+1)=0$

$x=-2,-1$

Clearly, $f^{\prime }\left(x\right)<0$ if $x>-1$ & $x<-2$

$f^{\prime }\left(x\right)>0$ if $-2<x<-1$

Thus, $f\left(x\right)$ is increasing in $(-2,-1)$, decreasing in $(-\infty ,-2)\cup (-1,\infty )$.