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 critical points of the function $f\left(x\right)=2\ln \left(x\right)-x^2$.

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

A container is being filled with a liquid at a rate that changes over time. The volume of the liquid in the container after $t$ seconds is given by $V=200(50-t{)}^2$ cubic meters. At what time is the magnitude of the flow rate into the container a maximum?

A farmer is monitoring the growth of a crop over a $120$-day period. The height of the crop in centimeters is given by the function $h(t)=\begin{cases}\frac95t^2,\text{ if }0\le t<60\\ -\frac95\left(t^2-240t+120\right),\text{if }60\le t<120\end{cases}$ , where $t$ is the time in days since initially planting their crops. When is the growth rate of the crop at a maximum?

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.

Suppose the domain of $f\left(x\right)$ is $(-1,1)$. Determine the intervals where $f\left(x\right)$ is increasing and decreasing using the graph of its derivative $f^{\prime }\left(x\right)$.