Given the derivative $f^{\prime}(x)=8\cos x-4\sin2x$ on the interval $[0, 2\pi]$, identify the $x$-coordinates of the local maxima and minima of $f$, as well as the intervals where $f$ is increasing or decreasing.

• A. Critical points: $x = \frac{\pi}{2}, \frac{3\pi}{2}$,Intervals of increase: $\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$,Intervals of decrease: $(0,\frac{\pi }{2})$, $(\frac{3\pi }{2},2\pi )$,Local maximum: $x = \frac{\pi}{2}$,Local minimum: $x = \frac{3\pi}{2}$.
• B. Critical points: $x=\frac{\pi }{2},\frac{3\pi }{4}$,Intervals of increase: $(0,\frac{\pi }{2})$, $(\frac{3\pi }{2},2\pi )$,Intervals of decrease: $\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$,Local maximum: $x = \frac{\pi}{2}$,Local minimum: $x = \frac{3\pi}{2}$.
• C. Critical points: $x = \frac{\pi}{2}, \frac{3\pi}{2}$,Intervals of increase: $(0,\frac{\pi }{2})$, $(\frac{3\pi }{2},2\pi )$,Intervals of decrease: $\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$,Local maximum: $x = \frac{\pi}{2}$,Local minimum: $x = \frac{3\pi}{2}$.
• D. Critical points: $x = \frac{\pi}{2}, \frac{3\pi}{2}$,Intervals of increase: $(0,\frac{\pi }{2})$, $(\frac{3\pi }{2},2\pi )$,Intervals of decrease: $\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$,Local maximum: $x = \frac{3\pi}{2}$,Local minimum: $x = \frac{\pi}{2}$.

Answer: (C)