Question

Find the greatest and the least values of the following function.
$f\left(x\right)=-x^3+3x^2+5$ on the interval $[0,3]$.

Answer 1

$f\left(x\right)=-x^3+3x^2+5$

$f^{\prime }\left(x\right)=-3x^2+6x=0\Rightarrow x=0,-2$

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

$f\left(0\right)=f\left(3\right)=5theminimum,f(-2)=25themaximum$