 Solution

As in the diagram, an inscribed cylinder rests on $z=0$, is coaxial with the paraboloid, and has its top rim on the curved boundary. ThusAllowing contact with the boundary is the usual meaning of inscribed. Form the Lagrange multiplier functionAt a nondegenerate stationary cylinder, its two derivative equations areEliminating $\lambda$ gives $h=H{r}^2/R^2$. The constraint then gives $h=H/2$ and $r=R/\sqrt{2}$, soThe degenerate endpoints $r=0$ or $h=0$ have zero volume, consistent with the positive stationary value being the maximum. If the strict inequalities defining the open interior were interpreted as forbidding boundary contact, the same value would instead be a supremum approached by smaller cylinders.