Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=6+12x+3x^2-2x^3$

Answer 1

We have

$f\left(x\right)=6+12x+3x^2-2x^3$

$f\left(x\right)=0+12+6x-6x^2$

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

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

$x^2-x-2=0$

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

$x=2,-1$

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

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

Thus , $f\left(x\right)$ increases in $(-1,2)$, decreases in $(-\infty ,-1)\cup (2,\infty )$