Locate the absolute extrema of the given function on the closed interval .
A) absolute max: (0, 0); absolute min
B) absolute max: ; absolute min (0, 0)
C) absolute max: ; absolute min
D) absolute max: ; no absolute min
E) no absolute max; absolute min:

Question

Locate the absolute extrema of the given function on the closed interval  . 
A) absolute max: (0, 0); absolute min 
B) absolute max: ; absolute min (0, 0) 
C) absolute max: ; absolute min 
D) absolute max: ; no absolute min 
E) no absolute max; absolute min:

Quizplus · Accepted Answer

To find the absolute extrema, we need to evaluate the function at the critical points within the interval and the endpoints of the interval. Taking the derivative of the function, we get:

f'(x) = 3x^2 - 6x

Setting this equal to zero, we get:

3x(x-2) = 0

So the critical points are x=0 and x=2. Evaluating the function at these critical points, we get:

f(0) = 0
f(2) = 16

Evaluating the function at the endpoints of the interval, we get:

f(-1) = 27
f(3) = 27

So the absolute maximum value is 27, which occurs at x=-1 and x=3, and the absolute minimum value is 0, which occurs at x=0. Therefore, choice C is the best answer.

Locate the Absolute Extrema of the Given Function on the Closed