Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.
A) ; relative maximum value:
B) ; relative maximum value:
C) ; relative maximum value:

Question

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. ​  ​
A) ; relative maximum value: 
B) ; relative maximum value: 
C) ; relative maximum value:

Quizplus · Accepted Answer

To find the critical point(s), we set the derivative equal to 0 and solve for x:
f'(x) = 3x^2 - 8x = x(3x-8) = 0
x = 0 or x = 8/3 are the critical points.
To classify the nature of each critical point, we need to take the second derivative:
f''(x) = 6x - 8
At x = 0, f''(0) = -8, which is negative, so the critical point at x = 0 is a relative maximum.
At x = 8/3, f''(8/3) = 2, which is positive, so the critical point at x = 8/3 is a relative minimum.
Therefore, the best choice is B, which correctly identifies the critical point at x = 0 as a relative maximum. Choices A and C incorrectly identify the critical points at x = 0 and x = 8/3, respectively.

Find the Critical Point(s) of the Function