The volume of the largest possible right circular cylinder that can be in inscribed in a sphere of radius $=\sqrt{3}$ is
 

$$\begin{gathered} {\mathrm{r}}^2+\frac{{\mathrm{h}}^2}{4}=3 \\ \Rightarrow {\mathrm{r}}^2=3-\frac{{\mathrm{h}}^2}{4} \end{gathered}$$

volume of cylinder

$$\begin{gathered} \mathrm{V}={\mathrm{\pi r}}^2\mathrm{h} \\ \mathrm{V}=\mathrm{\pi }\left(3-\frac{{\mathrm{h}}^2}{4}\right)\mathrm{h} \\ =\mathrm{\pi }\left(3\mathrm{h}-\frac{{\mathrm{h}}^3}{4}\right) \\ {\mathrm{V}}^1=\mathrm{\pi }\left(3-\frac{3{\mathrm{h}}^2}{4}\right) \end{gathered}$$

V is min or max

$$\begin{gathered} \Rightarrow {\mathrm{V}}^1=0 \\ 3-\frac{3{\mathrm{h}}^2}{4}=0 \\ \Rightarrow {\mathrm{h}}^2=4\Rightarrow \mathrm{h}=2 \end{gathered}$$

volume $$\begin{gathered} \mathrm{V}=\mathrm{\pi }\left(3\mathrm{h}-\frac{{\mathrm{h}}^3}{4}\right) \\ =\mathrm{\pi }\left(6-\frac{8}{4}\right)=4\mathrm{\pi } \end{gathered}$$