Given the derivative $f^{\prime }\left(x\right)=8\cos x-4\sin 2x$ 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: $(\frac{\pi }{2},\frac{3\pi }{2})$, 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: $(\frac{\pi }{2},\frac{3\pi }{2})$, 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: $(\frac{\pi }{2},\frac{3\pi }{2})$, 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: $(\frac{\pi }{2},\frac{3\pi }{2})$, Local maximum: $x=\frac{3\pi }{2}$, Local minimum: $x=\frac{\pi }{2}$.

Answer: (C)