= Solution
The zero <level set> is especially simple:
$$
\boxed{f=0\iff x=0\quad\text{or}\quad y=0\quad\text{or}\quad x^2+y^2=1}.
$$
The two axes and the unit <circle> divide the <plane> into regions where the signs alternate. Inside the <circle>, $f$ has the opposite sign to $xy$; outside it has the same sign. Closed <level curves> surround the two interior <local minima> and two interior <local maxima>; the zero curves cross at the five <saddle points of a scalar function>. Farther out, the nonzero <level curves> are open within the displayed square.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2-q12-contours.png]
{title=Contours, stationary points and gradient directions of the quartic surface}
{description=Left: signed <level curves> and the exact zero set consisting of both axes and the unit <circle>. Right: normalized <gradient> directions, including arrows on the zero curves. The nine <stationary points> are marked in both panels.}
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