Find all relative extrema of the function Use the Second-Derivative Test when applicable.
A) The relative maximum is
B) The relative minimum is
C) The relative maximum is
D) The relative minimum is
E) The relative maximum is

Question

​Find all relative extrema of the function  Use the Second-Derivative Test when applicable.
A) ​The relative maximum is 
B) ​​The relative minimum is 
C) ​​The relative maximum is 
D) ​​​The relative minimum is 
E) ​​The relative maximum is

Quizplus · Accepted Answer

We need to find the critical points of the function and then use the Second-Derivative Test to determine their nature. 
Taking the first derivative of the function, we get:
f'(x) = 3x^2 - 6x
Setting this equal to zero and solving for x, we get:
3x(x - 2) = 0
x = 0 or x = 2
These are the critical points of the function.
Taking the second derivative of the function, we get:
f''(x) = 6x - 6
Now, we can evaluate f''(0) and f''(2) to determine their nature:
f''(0) = -6, which is negative. So, at x = 0, we have a relative maximum.
f''(2) = 6, which is positive. So, at x = 2, we have a relative minimum.
Therefore, the correct choice is C, which says that the relative maximum is at x = 0 and the relative minimum is at x = 2.

​Find All Relative Extrema of the Function Use the Second-Derivative

Find All Relative Extrema of the Function Use the Second-Derivative