$\text { Locate the absolute extrema of the given function on the closed interval } [ - 60,60 ] \text {. }$ $f ( x ) = \frac { 60 x } { x ^ { 2 } + 36 }$
A) absolute max: $( 0,0 )$; absolute min $( - 6 , - 5 )$
B) absolute max: $( 6,5 )$; absolute min $( 0,0 )$
C) absolute max: $( 6,5 )$; absolute $\min ( - 6 , - 5 )$
D) absolute max: $( 6,5 )$; no absolute min
E) no absolute max; absolute min: $( - 6 , - 5 )$

Question

$\text { Locate the absolute extrema of the given function on the closed interval } [ - 60,60 ] \text {. }$ $f ( x ) = \frac { 60 x } { x ^ { 2 } + 36 }$ 
A) absolute max: $( 0,0 )$; absolute min $( - 6 , - 5 )$
B) absolute max: $( 6,5 )$; absolute min $( 0,0 )$
C) absolute max: $( 6,5 )$; absolute $\min ( - 6 , - 5 )$
D) absolute max: $( 6,5 )$; no absolute min
E) no absolute max; absolute min: $( - 6 , - 5 )$

Quizplus · Accepted Answer

First, we find any critical points in the interval $[-60,60]$ by finding where the derivative of $f(x)$ is equal to $0$ or undefined:

\begin{align*}
    f(x) &= \frac{60x}{x^2+36} \
    f'(x) &= \frac{60(x^2+36) - 60x(2x)}{(x^2+36)^2} \
    &= \frac{-120x^2+2160}{(x^2+36)^2} \
    &= \frac{-120(x^2-18)}{(x^2+36)^2}
\end{align*}

$f'(x)$ is undefined when $x = \pm 6\sqrt{5}$. Notice that these values are outside of the interval $[-60,60]$.

$f'(x) = 0$ when $x = 0$.

Next, we check the endpoints of the interval:
\begin{align*}
    f(-60) &= \frac{-3600}{-3594} \approx -1.0022 \
    f(60) &= \frac{3600}{3594} \approx 1.0022 \
    f(0) &= 0
\end{align*}

Therefore, $f$ attains an absolute maximum at $x = 6$, and $f$ attains an absolute minimum at $x = -6$ (since $f$ is an even function). The respective values of these extrema are $f(6) = \frac{180}{17}$ and $f(-6) = -\frac{180}{17}$.

Thus, the answer is $\boxed{	extbf{(C)}}$.

Locate the absolute extrema of the given function on the closed interval [−60,60]. \text { Locate the absolute extrema of the given function on the closed interval } [ - 60,60 ] \text {. } Locate the absolute extrema of the given function on the closed interval [−60,60].

$\text { Locate the absolute extrema of the given function on the closed interval } [ - 60,60 ] \text {. }$