$\text { Locate the absolute extrema of the function } f ( x ) = \sin \pi x \text { on the closed interval }$ $\left[ 0 , \frac { 1 } { 3 } \right]$
A) The absolute minimum is 0 , and it occurs at the left endpoint $x = 0$. The absolute maximum is $\frac { \sqrt { 3 } } { 2 }$, and it occurs at the right endpoint $x = \frac { 1 } { 3 }$.
B) The absolute minimum is 0 , and it occurs at the right endpoint $x = \frac { 1 } { 3 }$.
The absolute maximum is $\frac { 1 } { 2 }$, and it occurs at the left endpoint $x = 0$.
C) The absolute minimum is 0 , and it occurs at the left endpoint $x = 0$. The absolute maximum is $\frac { 1 } { 2 }$, and it occurs at the right endpoint $x = \frac { 1 } { 3 }$.
D) The absolute minimum is 0 , and it occurs at the right endpoint $x = \frac { 1 } { 3 }$.
The absolute maximum is $\frac { \sqrt { 2 } } { 2 }$ and it occurs at the left endpoint $x = 0$.
E) The absolute minimum is 0 , and it occurs at the left endpoint $x = 0$.
The absolute maximum is $\frac { \sqrt { 2 } } { 2 }$, and it occurs at the right endpoint $x = \frac { 1 } { 3 }$.

Question

Quizplus · Accepted Answer

First, we need to find the critical points of the function in the given closed interval. These critical points are the endpoints, which are $x=0$ and $x=\frac{1}{3}$. Next, we evaluate the function at these points and find which one gives the highest and which one gives the lowest value. 
$f(0) = \sin(0) = 0$
$f(\frac{1}{3}) = \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$
Therefore, the absolute minimum is 0 and it occurs at the left endpoint $x=0$. The absolute maximum is $\frac{\sqrt{3}}{2}$ and it occurs at the right endpoint $x=\frac{1}{3}$. Choice A is the only one that matches this result.

Locate the absolute extrema of the function f(x)=sin⁡πx on the closed interval \text { Locate the absolute extrema of the function } f ( x ) = \sin \pi x \text { on the closed interval } Locate the absolute extrema of the function f(x)=sinπx on the closed interval

$\text { Locate the absolute extrema of the function } f ( x ) = \sin \pi x \text { on the closed interval }$