10. 1/2 points details my notes scalc9 3.1.052. find the absolute maximum and absolute minimum values of f…

10. 1/2 points details my notes scalc9 3.1.052. find the absolute maximum and absolute minimum values of f on the given interval. f(x)=x³ - 9x² + 8, -4, 8 absolute minimum value -200 absolute maximum value need help? read it submit answer 11. -/2 points details my notes scalc9 3.1.059. find the absolute maximum and absolute minimum values of

Answer

Explanation:

Step1: Find the derivative

Differentiate $f(x)=x^{3}-9x^{2}+8$ using the power - rule. The derivative $f^\prime(x)=3x^{2}-18x = 3x(x - 6)$.

Step2: Find the critical points

Set $f^\prime(x)=0$. So $3x(x - 6)=0$. The solutions are $x = 0$ and $x = 6$, which are the critical points.

Step3: Evaluate the function at critical points and endpoints

Evaluate $f(x)$ at $x=-4,0,6,8$.

$f(-4)=(-4)^{3}-9(-4)^{2}+8=-64 - 144+8=-200$.
$f(0)=0^{3}-9(0)^{2}+8 = 8$.
$f(6)=6^{3}-9(6)^{2}+8=216-324 + 8=-100$.
$f(8)=8^{3}-9(8)^{2}+8=512-576+8=-56$.