Question

Find the greatest and the least values of the following functions.
$f\left(x\right)=2x^3-3x^2-12x+30$ on the interval $[-3,3]$.

Answer 1

$f\left(x\right)=2x^3-3x^2-12x+30$

$f^{\prime }\left(x\right)=6x^2-6x-12=0\Rightarrow (x-2)(x+1)=0$

$x=2,-1$

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

$f^{\prime \prime }\left(2\right)=18>0$,Hence $f\left(2\right)=10$ is the minimum

$f^{\prime \prime }(-1)=-18>0$,Hence $f(-1)=37$ is the maximum