Identify the open intervals on which the function $y = 9 x - 18 \cos x , 0

Question

Identify the open intervals on which the function $y = 9 x - 18 \cos x , 0 < x < 2 \pi \text { is }$ increasing or decreasing. 
A) increasing on $\left( \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 } \right) ;$ decreasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 11 \pi } { 6 } , 2 \pi \right)$
B) increasing on $\left( 0 , \frac { 11 \pi } { 6 } \right)$ and $\left( \frac { 7 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$
C) increasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 11 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 } \right)$ 
D) increasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 5 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 7 \pi } { 6 } , \frac { 5 \pi } { 6 } \right)$
E) increasing on $\left( \frac { 7 \pi } { 6 } , \frac { 5 \pi } { 6 } \right)$ and $\left( \frac { 5 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \right)$

Quizplus · Accepted Answer

We can obtain the derivative of the given function $y=9x-18\cos x$ as follows: 
$$y' = 9 + 18\sin x $$ 
For $0<x<\pi$, $\sin x$ is increasing, which means that $y'$ is increasing. Therefore, $y$ is increasing. 
For $\pi<x<2\pi$, $\sin x$ is decreasing, which means that $y'$ is decreasing. Therefore, $y$ is decreasing.

Now we need to find the intervals where $y$ is increasing and decreasing.

For $0<x<\pi$, $y'$ is positive and increasing, which means that $y$ is increasing. Therefore, $y$ is increasing on $\left(0,\frac{7\pi}{6}ight)$ and $\left(\frac{11\pi}{6},2\piight)$.

For $\pi<x<2\pi$, $y'$ is negative and decreasing, which means that $y$ is decreasing. Therefore, $y$ is decreasing on $\left(\frac{7\pi}{6},\frac{11\pi}{6}ight)$.

Thus, the correct choice is $\boxed{	extbf{(C)}}$.

Identify the Open Intervals on Which the Function y=9x−18cos⁡x,0<x<2π is y = 9 x - 18 \cos x , 0 < x < 2 \pi \text { is }y=9x−18cosx,0<x<2π is

Identify the Open Intervals on Which the Function $y = 9 x - 18 \cos x , 0 < x < 2 \pi \text { is }$