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

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.

On $y=0$, the restricted function is $g(x)=x^4$, so $g^{\prime }(x)=4x^3$ changes from negative to positive at zero. HenceThe restricted second derivative vanishes there, but the quartic term determines the minimum. This does not contradict the two-dimensional saddle: values decrease along other directions, including the diagonal near the origin.

On $y=x$, the restricted function is $g(x)=2x^4-16x^2$. Its derivative is $g^{\prime }(x)=8x(x^2-4)$, so the turning points are $x=-2,0,2$. The restricted second derivative is $g^{\prime \prime }=24x^2-32$, negative at zero and positive at $\pm 2$. ThusThe corresponding points in the plane are $(0,0)$ and $(\pm 2,\pm 2)$. Along the diagonal the derivative is $f_x+f_y$, not a single arbitrary partial derivative; here $f_x=f_y$, so their zeros and sign changes agree with the restricted-curve calculation.