Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / F / 2 / c

If a differentiable function has a constrained local extremum at a point of a smooth surface, its derivative vanishes in every tangent direction; equivalently its gradient is normal to the surface. For a regular equation $F=0$ this means $\nabla V=\lambda \nabla F$ for some real multiplier.

First the box is centred at the origin: holding the other coordinates of a vertex of a polyhedron fixed and comparing the two possible coordinates $u,v$ in the first direction gives $u^2=v^2$, so the distinct extremes are opposites; the other two directions give the same conclusion. Write its positive half-lengths as $ar,bs,ct$, with $r^2+s^2+t^2=1$. Its volume is $8abcrst$. A degenerate box has zero volume, so a positive maximum is interior in $r,s,t>0$. Lagrange multipliers applied to $\log (rst)$ give $1/r=2\lambda r$, and the analogous equations imply $r=s=t=1/\sqrt{3}$. Alternatively the arithmetic–geometric mean inequality applied to $r^2,s^2,t^2$ proves global maximality. Thus