Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=x^3-6x^2+9x+15$

Answer 1

$f\left(x\right)=x^3-6x^2+9x+15$

$f^{\prime }\left(x\right)v=3x^2-12x+9$

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

$=3(x-1)(x-3)$

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

Drawing its sign scheme we have

$\therefore f\left(x\right)$ is increasing in $(-\infty ,1)\cup (3,\infty )$

similarly $f^{\prime }\left(x\right)<0$ for decreasing i.e $(1,3)$