= Solution
Here the <stationary point> is the origin, of value two. The <Taylor expansion> is
$$
h=2+\frac{x^2+y^2}{2}+\frac{x^3+y^3}{6}+O((|x|+|y|)^4).
$$
Thus nearby <level curves> are approximately concentric circles. Globally the <level curves> above two are closed around the unique <minimum>, symmetric under interchanging $x$ and $y$, but not under changing either coordinate's sign. The faster exponential growth for positive coordinates compresses the curves on their positive sides.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2-q12-contours.png]
{title=Exponential level curves and gradient directions for a minimum and a saddle}
{description=Both panels show $|x|,|y|\leq2$. Left: $e^x+e^y-x-y$ has nested <level curves> around its minimum of value two. Right: $e^x-e^y-x+y$ has saddle-shaped <level curves> around value zero. Arrows show normalized <gradient> directions near the origin.}
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