Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 3 / 6D (Variational Principles)

For the boundary constraint $g(x,y)=x^2+y^2-1=0$, the Lagrange multiplier equations for $z=xy$ areNeither coordinate can vanish at a stationary point. Substitution gives $4{\lambda }^2=1$. For $\lambda =1/2$, $x=y=\pm 1/\sqrt{2}$ and $z=1/2$; for $\lambda =-1/2$, $x=-y=\pm 1/\sqrt{2}$ and $z=-1/2$. The constraint circle is compact, its gradient is nowhere zero, and every extremum must occur among these stationary points. ThusThe paired signs in each expression are linked. The inequality $2|xy|\le x^2+y^2=1$ also confirms the global values.

Let $c>0$ denote the stated unit cost of cylindrical wall, and assume $V>0$. The circular cylinder and hemisphere give the volume constraint and total costOnly the walls and cap have specified costs, so no separate base cost is included. Eliminating $h$ giveswith $0<r\le (3V/(2\pi ){)}^{1/3}$ from $h\ge 0$. Now $C^{\prime }(r)=-2cV/r^2+(16/3)\pi cr$ vanishes at $r^3=3V/(8\pi )$. Moreover $C^{\prime \prime }(r)=4cV/r^3+(16/3)\pi c>0$ and the stationary point is feasible, so it is the unique global minimum. The optimal dimensions are