 Optimal cylinder for a reciprocal-square constraint

For $q,C>0$, minimize a cylinder's volume $V=\pi R^{2}H$ over $R,H>0$ subject to $q^2/R^2+{\pi }^2/H^2=C$. Set $u=q^2/R^2$ and $v={\pi }^2/H^2$. Then $u+v=C$ and $V={\pi }^2{q}^2/(u\sqrt{v})$. Minimizing volume is equivalent to maximizing $u^2(C-u)$ on $0<u<C$. Its unique interior maximum is at $u=2C/3$, $v=C/3$, with divergence of volume at both endpoints. Thus the unique global optimum has