Determine all critical points for the function.
-$$f ( x ) = x ^ { 3 } - 12 x + 1$$
A) x = -2, x = 0, and x = 2
B) x = 2
C) x = -2 and x = 2
D) x = -2

Question

Quizplus · Accepted Answer

To find critical points, we first take the derivative and set it equal to zero: 
$$f'(x) = 3x^2 - 12 = 3(x^2 - 4) = 0$$
Solving for x, we get x = ±2. These are the only critical points because the second derivative, f''(x) = 6x, is never zero. Therefore, the answer is choice C, x = -2 and x = 2.

Determine All Critical Points for the Function f(x)=x3−12x+1f ( x ) = x ^ { 3 } - 12 x + 1f(x)=x3−12x+1

Determine All Critical Points for the Function $f ( x ) = x ^ { 3 } - 12 x + 1$