Question

find the critical points of the following function on the given interval. identify the absolute maximum and absolute minimum values (if they exist)

$f(x)=3x^{3}-36x^{2}$ on $-1,9$

identify all the critical points on the given interval. select the correct choice below and, if necessary, fill in the answer box within your choice.

a. the critical point(s) occur(s) at $x=$

(use a comma to separate answers as needed.)

b. there are no critical points for $f$

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=3x^{3}-36x^{2}\) using the power rule \((x^n)^\prime = nx^{n - 1}\) is \(f^\prime(x)=9x^{2}-72x\).

Step2: Set the derivative equal to zero and solve for \(x\)

Set \(f^\prime(x)=0\), so \(9x^{2}-72x = 0\). Factor out \(9x\): \(9x(x - 8)=0\).
Then \(9x=0\) gives \(x = 0\), and \(x - 8=0\) gives \(x = 8\). Both \(x = 0\) and \(x = 8\) are in the interval \([-1,9]\).

Answer:

A. The critical point(s) occur(s) at \(x = 0,8\)