Let VAB be the cone of height $\mathrm{h}$, semi-vertical angle $\alpha$ and let $\mathrm{x}$ be the radius of the base of the cylinder $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{DC}$ which is inscribed in the coneVAB. Then $\mathrm{OO}^{\prime}$ height of the cylinder $=\mathrm{VO}-\mathrm{VO}^{\prime}=\mathrm{h}-\mathrm{x} \cot \alpha$
volume of the cylinder
$=\pi x^2(h-x \cot \alpha)\quad \ldots(i)$
$\frac{d v}{d x}=2 \pi x h-3 \pi x^2 \cot \alpha$
For maxima or minima $\mathrm{v}, \frac{\mathrm{dv}}{\mathrm{dx}}=0 \Rightarrow \mathrm{x}=\frac{2 \mathrm{~h}}{3} \tan \alpha$ Now, $\frac{\mathrm{d}^2 \mathrm{v}}{\mathrm{dx}^2}=2 \pi \mathrm{h}-6 \pi \mathrm{x} \cot \alpha \quad$ When $\mathrm{x}=\frac{2 \mathrm{~h}}{3} \tan \alpha$, we have $\frac{\mathrm{d}^2 \mathrm{v}}{\mathrm{dx^{2 }}}=\pi(2 \mathrm{~h}-4 \mathrm{~h})=-2 \pi \mathrm{h} < 0$
$\Rightarrow v$ is maximum, when $x=\frac{2 h}{3} \tan \alpha$ $\mathrm{OO}^{\prime} \mathrm{h}-\mathrm{x} \cot \alpha=\mathrm{h}-\frac{2 \mathrm{~h}}{3}=\frac{\mathrm{h}}{3}$
$\therefore \quad$ The max. vol. of the cylinder is, $\mathrm{v}=\frac{4}{27} \pi \mathrm{h}^3 \tan ^2 \alpha$.