Question

Find the greatest and the least values of the following functions.
$f\left(x\right)=3x^3-9x^2+2$ on the interval $[-1,4]$.

Answer 1

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

$f^{\prime }\left(x\right)=9x^2-18x=0$

$x=0,2$

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

$f^{\prime \prime }\left(0\right)=-18<0$ Hence $f\left(0\right)=2$ is maximum

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