Determine the intervals on which the function $f\left(x\right)={\sin }^2\left(x\right)$ is increasing or decreasing on the interval $[-\frac{\pi }{2},\frac{\pi }{2}]$.

Determine the intervals of increasing/decreasing and the critical points of the function using the graph of $f^{\prime }\left(x\right)$.

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.