Find the exact location of all absolute extrema of the function $f ( x ) = x ^ { 3 } - 12 x$ with domain $[ - 7,7 ]$ .
Select all correct answers.

A) $( - 7 , - 259 )$ - absolute minimum
B) $( 7,259 )$ - absolute maximum
C) $( - 7,259 )$ - absolute maximum
D) $( 7 , - 259 )$ - absolute minimum
E) $( 0,259 )$ - absolute maximum

Question

Find the exact location of all absolute extrema of the function $f ( x ) = x ^ { 3 } - 12 x$ with domain $[ - 7,7 ]$ . 
Select all correct answers.
 
A) $( - 7 , - 259 )$ - absolute minimum
B) $( 7,259 )$ - absolute maximum
C) $( - 7,259 )$ - absolute maximum
D) $( 7 , - 259 )$ - absolute minimum
E) $( 0,259 )$ - absolute maximum

Quizplus · Accepted Answer

The function is a cubic polynomial, so it has at most two extrema. The derivative is $f'(x) = 3x^2 - 12$, which is zero at $x = \pm 2$. Since the second derivative is $f''(x) = 6x$, the function has an absolute minimum at $x = -2$ and an absolute maximum at $x = 2$. Since these values are within the domain, the absolute extrema are $(-7, -259)$ and $(7, 259)$.

Find the Exact Location of All Absolute Extrema of the Function