Question

determine the location and value of the absolute extreme values of f on the given interval, if they exist.
f(x)=x² - 8 on -2,4
what is/are the absolute maximum/maxima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute maximum/maxima is/are □ at x=□
(use a comma to separate answers as needed.)
b. there is no absolute maximum of f on the given interval.

Explanation:

Step1: Find the derivative of the function

The function is \(f(x)=x^{2}-8\). The derivative \(f^{\prime}(x) = 2x\).

Step2: Find the critical points

Set \(f^{\prime}(x)=0\), so \(2x = 0\), which gives \(x = 0\). The critical point \(x = 0\) lies within the interval \([-2,4]\).

Step3: Evaluate the function at the critical point and endpoints

• At \(x=-2\): \(f(-2)=(-2)^{2}-8=4 - 8=-4\).
• At \(x = 0\): \(f(0)=0^{2}-8=-8\).
• At \(x = 4\): \(f(4)=4^{2}-8=16 - 8 = 8\).

Answer:

A. The absolute maximum/maxima is/are at \(x = 4\)