Find the relative extrema, if any, of the function. Use the second derivative test, if applicable.
A) Relative minimum: f (- 1) = 17
B) Relative maximum: f (- 2) = 12
C) Relative maximum: f (- 1) = 17
D) Relative minimum: f (- 2) = 12
E) none

Question

Find the relative extrema, if any, of the function. Use the second derivative test, if applicable. 
A) Relative minimum: f (- 1) = 17 
B) Relative maximum: f (- 2) = 12 
C) Relative maximum: f (- 1) = 17 
D) Relative minimum: f (- 2) = 12 
E) none

Quizplus · Accepted Answer

By looking at the function, we can see that it is increasing to the left of x=-2 and decreasing to the right of x=-2. Therefore, f(-2) is a relative maximum. Similarly, the function is decreasing to the left of x=-1 and increasing to the right of x=-1. Therefore, f(-1) is a relative minimum. We can confirm this using the second derivative test:
f'(x) = 3x^2 - 4x
f''(x) = 6x - 4
f''(-2) = 6(-2) - 4 < 0, so f(-2) is a relative maximum.
f''(-1) = 6(-1) - 4 > 0, so f(-1) is a relative minimum.

Find the Relative Extrema, If Any, of the Function