$\text { Locate the absolute extrema of the function } y = x ^ { 2 } - 2 \cos x \text { on the closed interval }$ $[ - 3,5 ] \text {. Round your answers to four decimal places. }$
A) left endpoint: $( - 3,10.9800 )$
right endpoint: $( 5,24.4327 )$ absolute maximum
critical points: $( 0,0.0000 )$ absolute minimum

B) left endpoint: $( - 3 , - 6.2822 )$ absolute minimum
right endpoint: $( 5,8.0822 )$ absolute maximum
critical points: none

C) left endpoint: $( - 3 , - 6.2822 )$ absolute minimum
right endpoint: $( 5,8.0822 )$ absolute maximum
critical points: $( 0,0.0000 )$

D) left endpoint: $( - 3,10.9800 )$ absolute minimum
right endpoint: $( 5,24.4327 )$ absolute maximum
critical points: none

E) left endpoint: $( - 3,10.9800 )$
right endpoint: $( 5,24.4327 )$ absolute maximum
critical points: $( 0 , - 2.0000 )$ absolute minimum

Question

$\text { Locate the absolute extrema of the function } y = x ^ { 2 } - 2 \cos x \text { on the closed interval }$ $[ - 3,5 ] \text {. Round your answers to four decimal places. }$ 
A) left endpoint: $( - 3,10.9800 )$
right endpoint: $( 5,24.4327 )$ absolute maximum 
critical points: $( 0,0.0000 )$ absolute minimum

B) left endpoint: $( - 3 , - 6.2822 )$ absolute minimum
 right endpoint: $( 5,8.0822 )$ absolute maximum
 critical points: none

C) left endpoint: $( - 3 , - 6.2822 )$ absolute minimum
 right endpoint: $( 5,8.0822 )$ absolute maximum
 critical points: $( 0,0.0000 )$

D) left endpoint: $( - 3,10.9800 )$ absolute minimum
 right endpoint: $( 5,24.4327 )$ absolute maximum
 critical points: none

E) left endpoint: $( - 3,10.9800 )$
right endpoint: $( 5,24.4327 )$ absolute maximum
 critical points: $( 0 , - 2.0000 )$ absolute minimum

Quizplus · Accepted Answer

To find the absolute extrema of the function $y = x^2 - 2\cos(x)$ on the interval $[-3,5]$, we first evaluate the function at the endpoints and then find its critical points by taking the derivative and setting it equal to zero. The derivative is $y' = 2x + 2\sin(x)$. Setting $y' = 0$ gives critical points, one of which is at $x = 0$, leading to $y(0) = -2$, which is the absolute minimum within the interval. Evaluating the function at the endpoints, $y(-3) = 9 - 2\cos(-3) \approx 10.9800$ and $y(5) = 25 - 2\cos(5) \approx 24.4327$, confirms the absolute maximum at the right endpoint.

Locate the absolute extrema of the function y=x2−2cos⁡x on the closed interval \text { Locate the absolute extrema of the function } y = x ^ { 2 } - 2 \cos x \text { on the closed interval } Locate the absolute extrema of the function y=x2−2cosx on the closed interval

$\text { Locate the absolute extrema of the function } y = x ^ { 2 } - 2 \cos x \text { on the closed interval }$