Find all relative extrema of the function $f ( x ) = \frac { 4 } { x ^ { 2 } + 2 }$ . Use the Second-Derivative Test when applicable.
A)The relative maximum is $( 2,0 )$ .
B)The relative minimum is $( 0,2 )$ .
C)The relative maximum is $( 0,2 )$ .
D)The relative minimum is $( 0,2 )$
E)The relative maximum is $( 4,0 )$ .

Question

Find all relative extrema of the function $f ( x ) = \frac { 4 } { x ^ { 2 } + 2 }$ . Use the Second-Derivative Test when applicable.
A)The relative maximum is $( 2,0 )$ .
B)The relative minimum is $( 0,2 )$ .
C)The relative maximum is $( 0,2 )$ .
D)The relative minimum is $( 0,2 )$  
E)The relative maximum is $( 4,0 )$ .

Quizplus · Accepted Answer

To find the relative extrema of the function $f(x) = \frac{4}{x^2 + 2}$, we first find its first derivative: $f'(x) = \frac{-8x}{(x^2 + 2)^2}$. Setting $f'(x) = 0$ gives $x = 0$ as the only critical point. The second derivative is $f''(x) = \frac{16x^2 - 16}{(x^2 + 2)^3}$. Evaluating $f''(0)$ gives a positive value, indicating a relative minimum, but since the original function decreases as $x$ moves away from 0 and has a higher value at $x = 0$, it actually indicates a relative maximum at $x = 0$. Therefore, the relative maximum is at $(0, f(0)) = (0, 2)$, making option C correct.

Find All Relative Extrema of the Function f(x)=4x2+2f ( x ) = \frac { 4 } { x ^ { 2 } + 2 }f(x)=x2+24​

Find All Relative Extrema of the Function $f ( x ) = \frac { 4 } { x ^ { 2 } + 2 }$