Question

Find the greatest and the least values of the following functions.
$f\left(x\right)=x^4-8x^2-9$ on the interval [0, 3].

Answer 1

$f\left(x\right)=x^4-8x^2-9$

$f^{\prime }\left(x\right)=4x^3-16x=0\Rightarrow x=-2,0,2$

$f^{\prime \prime }\left(x\right)=12x^2-16$

At $x=0,f^{\prime \prime }\left(x\right)<0$ hence f(x) is maximum here

At $x=2,f^{\prime \prime }\left(x\right)>0$ hence f(x) is minimum here

$f\left(0\right)=-9ismaximum,f\left(2\right)=-25isminimum$