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.

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: $x=\frac{3\pi }{2}$.

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: $x=\frac{3\pi }{2}$.

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: $x=\frac{3\pi }{2}$.

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: $x=\frac{\pi }{2}$.