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

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 $f^{\prime}(x)=8\cos x-4\sin2x$ 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.