Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=2x^3-12x^2+18x+15$

Answer 1

$f\left(x\right)=2x^3-12x^2+18x+15$

$f^{\prime }\left(x\right)=6x^2-24x+18$

$6x^2-24x+18=0$

$x^2-4x+3=0$

$x^2-3x-x+3=0$

$x=3,x=1$

The points $x=1,3$ divided the real line into three intervals

$(-\infty ,1),(1,3),(3,\infty )$

In interval $(-\infty ,1)f^{\prime }\left(x\right)>0$

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

In interval $(1,3)f^{\prime }\left(x\right)<0$

$f\left(x\right)$ is decreasing in $(1,3)$

In the interval $(3,\infty )f^{\prime }\left(x\right)>0$

$f\left(x\right)$ increasing in $(3,\infty )$.