Find the absolute extreme values of the function on the interval.
-$$f ( x ) = 5 x ^ { 4 / 3 } , - 27 \leq x \leq 8$$
A) absolute maximum is 405 at x = -27 ; absolute minimum is 80 at x = 8
B) absolute maximum is 80 at x = 8 ; absolute minimum is 0 at x = 0
C) absolute maximum is 81 at x = -27 ; absolute minimum is 0 at x = 0
D) absolute maximum is 405 at x = -27 ; absolute minimum is 0 at x = 0

Question

Quizplus · Accepted Answer

To find the absolute extreme values of the function, we first take the derivative:
$$f'(x) = \frac{20}{3} x^{1/3}$$
Notice that the derivative is undefined at $x=0$ but both $-27$ and $8$ are within the given interval. 
The critical point occurs when $f'(x) = 0$, which is when $x=0$. However, this is not within the interval, so we only need to check the endpoints $x=-27$ and $x=8$ as well as the critical point $x=0$ (since it's a critical point in the domain of the function, even though it's not in the interval). 
\begin{align*}
f(-27) &= 5(-27)^{4/3}=405 \
f(0) &= 5(0)^{4/3}=0 \
f(8) &= 5(8)^{4/3}=80
\end{align*}
Therefore, the absolute maximum is $405$ at $x=-27$ and the absolute minimum is $0$ at $x=0$, so the answer is D.

Find the Absolute Extreme Values of the Function on the Interval