Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 6A / b / Solution

The stationary equations areThus $y=-x$ and $4x(x^2-2)=0$, givingThe Hessian matrix isAt either nonzero stationary point,so both are strict local minima.

Along a line through the origin,whereWhen $\cos \gamma \ne 0$ and $r=\tan \gamma$,Consequently the origin is a local minimum along the line whenincluding the equality cases because the positive quartic term then leads. It is also a minimum on the vertical line. It is a local maximum along the line whenIn the first case the graph is locally bowl-shaped. In the second it initially bends downward from the origin, but the positive quartic term eventually turns it upward, producing the usual double-well profile.