Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 12 / d

The gradient components are $f_x=2x(1+6x^2)$ and $f_y=2y(1-6y^2)$. Their real simultaneous zeros are $(0,0)$ and $(0,\pm 1/\sqrt{6})$. The Hessian matrix is diagonal:At the origin it is positive definite; at each other critical point it is $\mathrm{diag}(2,-4)$ and indefinite. HenceThere are no local maxima. The contours are symmetric under either coordinate reflection. Small positive levels have a closed component surrounding the minimum; these meet the saddle-level separatrices at level $1/12$. That critical contour has the useful exact form $3x^4+x^2=3(y^2-1/6{)}^2$. Other components extend toward large $|y|$, where the $-3y^4$ term makes the function unbounded below.