Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 17 / a / iv / Solution

The interior values satisfy $-1/27<f<0$ away from the minimum. The contours are closed curves surrounding $(1/3,-1/3)$, approximately ellipses near that point, and approach the three edges as the level tends to zero. To see their shape globally, put $u=x$, $v=-y$, $w=1-x+y$. These are positive inside and satisfy $u+v+w=1$, while $f=-uvw$. Since $\log (uvw)=\log u+\log v+\log w$ is strictly concave, its nonempty superlevel sets are convex and surround the unique maximum of $uvw$. Thus the negative contours nest without additional components, as in the figure.