Find the absolute extreme values of the function on the interval.
-$$f ( x ) = 6 x ^ { 2 / 3 } , - 27 \leq x \leq 8$$
A) absolute maximum is 24 at $x = 8$; absolute minimum is 0 at $x = 0$
B) absolute maximum is 54 at $x = - 27$; absolute minimum is 0 at $x = 0$
C) absolute maximum is 54 at $x = - 27$; absolute minimum is 24 at $x = 8$
D) absolute maximum is 9 at $x = - 27$; absolute minimum is 0 at $x = 0$

Question

Quizplus · Accepted Answer

To find the absolute extreme values, we evaluate the function at the endpoints and any critical points within the interval. The function is $f(x) = 6x^{2/3}$. At $x = -27$, $f(-27) = 6(-27)^{2/3} = 54$. At $x = 8$, $f(8) = 6(8)^{2/3} = 24$. The function has no critical points where its derivative is undefined or zero within the interval. Therefore, the absolute maximum is 54 at $x = -27$, and the absolute minimum is 0 at $x = 0$, since $f(0) = 6(0)^{2/3} = 0$.

Find the Absolute Extreme Values of the Function on the Interval