= 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 $-32$.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013-resit-quartic-contours-and-restrictions.png]
{title=Quartic contours on the required square and one-dimensional restrictions near their turning points}
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.
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