Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 11 / c / Solution

The contours have exchange symmetry $f(x,y)=f(y,x)$ and inversion symmetry $f(-x,-y)=f(x,y)$. Negative levels between $-32$ and zero form separate loops around the two minima. The zero contour meets itself at the origin, reflecting the saddle there; positive levels form a single outer closed contour. At large radius the positive quartic terms dominate, so the contours remain bounded. There are no contours below $-32$.

The diagram marks the two minima and central saddle. The restrictions are classified separately below: a minimum or maximum along a single line need not be a two-dimensional extremum.