Question

Find the greatest and the least values of the following function.
$f\left(x\right)=2x^3-9x^2+12x$ on the interval $[0,3]$.

Answer 1

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

$f^{\prime }\left(x\right)=6x^2-18x+12=0\Rightarrow (x-1)(x-2)=0$

$x=1,2$

$f^{\prime \prime }\left(x\right)=(2x-3)6$

$f^{\prime \prime }\left(1\right)=-6<0$ hence $f\left(1\right)=5$ is maximum

$f^{\prime \prime }\left(2\right)=6>0$ hence $f\left(2\right)=4$ is minimum