Locate the absolute extrema of the function on the closed interval [0,6].
A)absolute max: (6, 144) ; absolute min: (2, -16)
B)absolute max: (2, -16) ; absolute min: (6, 144)
C)absolute max: (6, 144) ; no absolute min
D)no absolute max; Absolute min: (6, 144)
E)no absolute max or min

Question

Locate the absolute extrema of the function  on the closed interval [0,6].
A)absolute max: (6, 144) ; absolute min: (2, -16) 
B)absolute max: (2, -16) ; absolute min: (6, 144) 
C)absolute max: (6, 144) ; no absolute min 
D)no absolute max; Absolute min: (6, 144) 
E)no absolute max or min

Quizplus · Accepted Answer

To find the absolute extrema of a function on a closed interval, we need to evaluate the function at the endpoints (0 and 6) and at any critical points in between. To find the critical points, we take the derivative and set it equal to zero:

f'(x) = 3x^2 - 36x + 54 
f'(x) = 3(x^2 - 12x + 18) 
f'(x) = 3(x - 6)(x - 3)

The critical points are x = 3 and x = 6. Evaluating the function at these points and the endpoints, we get:

f(0) = 0 
f(3) = -16 
f(6) = 144

Therefore, the absolute minimum occurs at (2, -16) and the absolute maximum occurs at (6, 144). Choice A is the only one that lists these as the correct answers.

Locate the Absolute Extrema of the Function on the Closed