Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 20 / b

Solve the Lagrange multiplier equations by separating cases according to which coordinates vanish, so that division by a coordinate loses no stationary points.

If $x\ne 0$ and $y=z=0$, the surface constraint givesIf $x\ne 0$ and exactly one of $y,z$ is nonzero, the equation for that nonzero coordinate forces $\lambda =0$, while the $x$ equation gives $\lambda =2a^2{x}^2>0$. Hence there are no stationary points in that case.

If $x=0$ and $y,z\ne 0$, the two remaining equations give $\lambda =a^2{z}^2=b^2{y}^2$. The surface constraint therefore has equal contributions from $y$ and $z$, giving four stationary points:If all three coordinates are nonzero, the equations implyso $z^2=2x^2$ and $y^2=2a^2{x}^2/b^2$. Substitution in the surface constraint gives, with $D=4a^2+b^2$,Thus the remaining eight stationary points and their temperatures areFor example, their common temperature follows from $x^4+y^2{z}^2=a^4{b}^4/D^2+4a^6{b}^2/D^2=a^4{b}^2/D$. The cases with $x=0$ and $yz=0$ are exactly the coldest points already listed. This exhausts all possibilities, giving fourteen other stationary points in addition to those four.