Find the absolute extreme values of the function on the interval.
-$$g ( x ) = 9 - 8 x ^ { 2 } , - 2 \leq x \leq 3$$
A) absolute maximum is 9 at $x = 0$; absolute minimum is $- 63$ at $x = 3$
B) absolute maximum is 8 at $x = 0$; absolute minimum is $- 81$ at $x = 3$
C) absolute maximum is 72 at $x = 0$; absolute minimum is $- 23$ at $x = - 2$
D) absolute maximum is 18 at $x = 0$; absolute minimum is $- 23$ at $x = 3$

Question

Quizplus · Accepted Answer

To find the absolute extreme values of a function, we need to find the critical values (where the derivative is zero or undefined) and the endpoints of the interval. Taking the derivative of $g(x)$, we get $g'(x) = -16x$. The derivative is zero at $x=0$ and undefined at neither endpoint. Next, we evaluate $g(x)$ at $x=-2$, $x=0$, and $x=3$, which gives us $g(-2) = 9 - 8(-2)^2 = -55$, $g(0) = 9 - 8(0)^2 = 9$, and $g(3) = 9 - 8(3)^2 = -63$. Therefore, the absolute maximum is 9 at $x=0$ and the absolute minimum is $-63$ at $x=3$. Choice A correctly matches these values.

Find the Absolute Extreme Values of the Function on the Interval