Find the relative extrema, if any, of the function.Use the second derivative test, if applicable.

Question

Quizplus · Accepted Answer

The function has two critical points, -2 and -1. To determine the relative extrema, we need to use either the first or second derivative test.

Taking the first derivative and setting it equal to zero, we get:
f'(x) = 3x^2 + 4x - 8
3x^2 + 4x - 8 = 0
(3x-4)(x+2) = 0
x = -2 or x = 4/3

So, we have two critical points, -2 and 4/3.

Now, we take the second derivative:
f''(x) = 6x + 4

Plug in the critical points:
f''(-2) = -8  < 0, so f(-2) is a relative maximum
f''(4/3) = 12 > 0, so f(4/3) is a relative minimum.

Therefore, the answer is D, as the function has a relative minimum at x = 4/3 and a relative maximum at x = -2.

Find the Relative Extrema, If Any, of the Function