= Maximal cylinder and box in an elliptic paraboloid
{title2=$V_{\rm cylinder}=\pi R^2H/4,\quad V_{\rm box}=R^2H/2$}
For the rotational <elliptic paraboloid> $z=H(1-(x^2+y^2)/R^2)$ above $z=0$, allow inscribed shapes to touch the boundary. A coaxial <cylinder> of radius $r$ and height $h$ obeys $h+Hr^2/R^2=H$. Applying a <Lagrange multiplier> to $\pi r^2h$ gives $h=Hr^2/R^2$, hence $h=H/2$ and $r=R/\sqrt2$.
A centred <rectangular parallelepiped> with horizontal side lengths $a,b$ and height $c$ obeys $c+H(a^2+b^2)/(4R^2)=H$. The <Lagrange multiplier> equations for $abc$ give $a=b$ and $c=Ha^2/(2R^2)$, hence $a=b=R,c=H/2$. These yield the displayed maximal volumes; both positive stationary shapes are interior optima among the dimension variables, while their degenerate boundary shapes have zero volume. The cylinder's volume is $\pi/2$ times the box's. If strict containment in the open region is required, these are suprema approached by slightly smaller shapes rather than attained maxima.
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