Find all relative extrema of the function $f ( x ) = \sqrt { 36 - x ^ { 2 } }$ . Use the Second-Derivative Test when applicable.
A)The relative minimum is $( 0,6 )$ and the relative maximum is $( - 6,0 )$ .
B)The relative maximum is $( 0,6 )$ .
C)The relative minimum is $( 0,6 )$ .
D)The relative maximum is $( 0,6 )$ and the relative minima are $( 6,0 )$ and $( - 6,0 )$ .
E)The relative minimum is $( 0,6 )$ and the relative maximum is $( 6,0 )$ .

Question

Quizplus · Accepted Answer

The function $f(x) = \sqrt{36 - x^2}$ reaches its maximum value when the expression under the square root is maximized, which occurs at $x = 0$, giving $f(0) = \sqrt{36} = 6$. This is a relative (and absolute) maximum because the function value decreases as $x$ moves away from 0 in either direction. There are no relative minima within the domain of the function because the function approaches 0 at $x = \pm 6$ but does not have a value less than 0 within its domain, making $(0,6)$ the only relative extremum as a maximum.

Find All Relative Extrema of the Function f(x)=36−x2f ( x ) = \sqrt { 36 - x ^ { 2 } }f(x)=36−x2​

Find All Relative Extrema of the Function $f ( x ) = \sqrt { 36 - x ^ { 2 } }$