The height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is

Let $r$ be the base radius and $h$  be the height of cylinder

$$\begin{gathered} R^2=\frac{h^2}{4}+r^2 \\ v=\pi r^{2}h \\ v=\pi (R^2-\frac{h^2}{4})h \\ v=\pi R^{2}h-\frac{h^3}{4} \\ \frac{dv}{dh}=R^2\mathrm{\pi }-\frac{3{\mathrm{h}}^2}{4} \end{gathered}$$

For max or min   $\frac{dv}{dh}=0\Rightarrow h=\frac{2R}{\sqrt{3}}$