Given the derivative $f^{\(\prime\)}(x)=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: $\(\left\)(\(\frac{\pi}{2}\), \(\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 = \(\frac{3\pi}{2}\)$.

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 = \(\frac{3\pi}{2}\)$.

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 = \(\frac{3\pi}{2}\)$.

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 = \(\frac{\pi}{2}\)$.