= Solution
Since $f>0$, a level $f=c$ can occur only for $c>0$. Taking its <natural logarithm> gives $xy=\ln c$. Thus $c>1$ produces rectangular <hyperbolas> in the first and third quadrants, $0<c<1$ produces them in the second and fourth quadrants, and $c=1$ consists of both coordinate axes.
The <gradient> $e^{xy}(y,x)$ vanishes only at $(0,0)$, where $f=1$. The <Hessian matrix> there is $\begin{pmatrix}0&1\\1&0\end{pmatrix}$, with negative <determinant>. Equivalently, $f(x,x)>1$ while $f(x,-x)<1$ for nonzero small $x$. Hence the only <stationary point> is a \b[<saddle point> at $(0,0,1)$] on the graph; its location in the contour <plane> is the origin.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2-exponential-contours.png]
{title=Contours of the exponential of xy, showing the unique saddle at the origin}
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