Use a graphing utility to graph the function over the indicated interval and approximate any local maxima and local
minima. Determine where the function is increasing and where it is decreasing. If necessary, round answers to two
decimal places.
-$$f ( x ) = x ^ { 3 } - 12 x + 2 ; ( - 5,5 )$$
A) local maximum at $( - 2,18 )$
local minimum at $( 0,0 )$
local minimum at $( 2 , - 14 )$

B) local minimum at $( 0,0 )$

C) local maximum at $( - 2,18 )$
local minimum at $( 2 , - 14 )$

D) none

Question

Use a graphing utility to graph the function over the indicated interval and approximate any local maxima and local
minima. Determine where the function is increasing and where it is decreasing. If necessary, round answers to two
decimal places.
-$$f ( x ) = x ^ { 3 } - 12 x + 2 ; ( - 5,5 )$$
A) local maximum at $( - 2,18 )$
local minimum at $( 0,0 )$
local minimum at $( 2 , - 14 )$

B) local minimum at $( 0,0 )$

C) local maximum at $( - 2,18 )$
local minimum at $( 2 , - 14 )$

D) none

Quizplus · Accepted Answer

To find local maxima and minima, we first find the derivative of the function $f'(x) = 3x^2 - 12$. Setting this equal to zero gives $x^2 = 4$, so $x = \pm 2$. Testing these values in the second derivative, $f''(x) = 6x$, we find that $x = -2$ gives a positive value (indicating a local minimum) and $x = 2$ gives a negative value (indicating a local maximum), which is actually the opposite of the usual interpretation of the second derivative test. The correct interpretation is that a positive second derivative at a point indicates concavity upwards and thus a local minimum, while a negative second derivative indicates concavity downwards and thus a local maximum. However, in this context, the correct identification comes from evaluating the original function at these points: $f(-2) = -2^3 - 12(-2) + 2 = 18$ and $f(2) = 2^3 - 12(2) + 2 = -14$, showing that $(-2, 18)$ is a local maximum and $(2, -14)$ is a local minimum. There is no local minimum at $(0,0)$ because the function does not have a turning point there.

Use a Graphing Utility to Graph the Function Over the Indicated