Use a graphing utility to graph the function over the indicated interval and approximate any local maxima and local
minima. If necessary, round answers to two decimal places.
-$$f ( x ) = x ^ { 3 } - 12 x + 2 ; ( - 5,5 )$$
A) local minimum at (0, 0)
B) local maximum at (-2, 18) local minimum at (0, 0)
Local minimum at (2, -14)
C) local maximum at (-2, 18)
D) none local minimum at (2, -14)

Question

Quizplus · Accepted Answer

To find local maxima and minima, we first find the derivative of the function $f(x) = x^3 - 12x + 2$, which is $f'(x) = 3x^2 - 12$. Setting the derivative equal to zero gives $3x^2 - 12 = 0$, or $x^2 = 4$, so $x = \pm 2$. Evaluating $f(x)$ at these points gives $f(-2) = (-2)^3 - 12(-2) + 2 = -8 + 24 + 2 = 18$ and $f(2) = (2)^3 - 12(2) + 2 = 8 - 24 + 2 = -14$. Thus, there is a local maximum at $(-2, 18)$ and a local minimum at $(2, -14)$, matching option C.

Use a Graphing Utility to Graph the Function Over the Indicated