$\text { Find the relative maxima of } f ( x ) = \cos ^ { 2 } ( 4 x ) \text { on the interval } ( 0,3.142 ) \text { by applying }$
the First Derivative Test. Round numerical values in your answer to three decimal places.
A) relative maxima: $( 1.571,1 ) , ( 1.963,1 ) , ( 2.356,13.142 )$
B) relative maxima: $( 0.393,0 ) ( 1.178,0 ) , ( 1.963,0 ) , ( 2.749,0 )$
C) relative maxima: $( 0.785,1 ) , ( 1.571,1 ) , ( 2.356,1 )$
D) relative maxima: $( 0.393,1 ) , ( 1.178,1 ) , ( 1.963,1 ) , ( 2.749,1 )$
E) relative maxima: $( 0.393,1 ) , ( 0.785,1 ) , ( 1.178,1 )$

Question

$\text { Find the relative maxima of } f ( x ) = \cos ^ { 2 } ( 4 x ) \text { on the interval } ( 0,3.142 ) \text { by applying }$ 
the First Derivative Test. Round numerical values in your answer to three decimal places. 
A) relative maxima: $( 1.571,1 ) , ( 1.963,1 ) , ( 2.356,13.142 )$
B) relative maxima: $( 0.393,0 ) ( 1.178,0 ) , ( 1.963,0 ) , ( 2.749,0 )$
C) relative maxima: $( 0.785,1 ) , ( 1.571,1 ) , ( 2.356,1 )$
D) relative maxima: $( 0.393,1 ) , ( 1.178,1 ) , ( 1.963,1 ) , ( 2.749,1 )$
E) relative maxima: $( 0.393,1 ) , ( 0.785,1 ) , ( 1.178,1 )$

Quizplus · Accepted Answer

To find the relative maxima of $f(x) = \cos^2(4x)$, we first find its derivative, set it to zero, and solve for $x$. The derivative of $f(x)$ with respect to $x$ is $f'(x) = -8\sin(4x)\cos(4x) = -4\sin(8x)$. Setting $f'(x) = 0$ gives $\sin(8x) = 0$, which implies $8x = n\pi$, where $n$ is an integer. For $x$ in the interval $(0, 3.142)$, the solutions are $x = \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}$, which approximate to $0.785, 1.571, 2.356$, respectively. At these points, $\cos^2(4x) = 1$, since $\cos(4x)$ is either $1$ or $-1$, and squaring these values gives $1$. These points are relative maxima because the function value changes from increasing to decreasing at these points, as confirmed by the First Derivative Test.

Find the relative maxima of f(x)=cos⁡2(4x) on the interval (0,3.142) by applying \text { Find the relative maxima of } f ( x ) = \cos ^ { 2 } ( 4 x ) \text { on the interval } ( 0,3.142 ) \text { by applying } Find the relative maxima of f(x)=cos2(4x) on the interval (0,3.142) by applying

$\text { Find the relative maxima of } f ( x ) = \cos ^ { 2 } ( 4 x ) \text { on the interval } ( 0,3.142 ) \text { by applying }$