Consider the function $$y = \sin \frac { x } { 4 } - 5$$ on the interval $$[ 0,8 \pi ]$$ . Specify the intervals in which the function is increasing.
A) $$( 0,2 \pi )$$
B) $$( 0,2 \pi ) , ( 6 \pi , 8 \pi )$$
C) $$( 6 \pi , 8 \pi )$$
D) $$( 2 \pi , 6 \pi ) , ( 8 \pi , 9 \pi )$$
E) $$( 0,8 \pi )$$

Question

Quizplus · Accepted Answer

The derivative of the function is $\frac 14 \cos \frac{x}{4}$.

In the interval $[0, 2\pi]$, the cosine is positive, so the derivative is positive and the function is increasing.

In the interval $[2\pi, 4\pi]$, the cosine is negative, so the derivative is negative and the function is decreasing.

In the interval $[4\pi, 6\pi]$, the cosine is positive again, so the function is increasing.

In the interval $[6\pi, 8\pi]$, the cosine is negative, so the function is decreasing.

Thus, the function is increasing on intervals $(0,2\pi)$ and $(6\pi,8\pi)$.

Consider the Function y=sin⁡x4−5y = \sin \frac { x } { 4 } - 5y=sin4x​−5

Consider the Function $y = \sin \frac { x } { 4 } - 5$