= Solution
The sign of $f$ is the sign of $1-4xy$. Therefore
$$
\boxed{f>0\text{ when }xy<\frac14,\qquad f<0\text{ when }xy>\frac14.}
$$
The positive region includes both coordinate axes, the second and fourth quadrants, and the portions of the first and third quadrants on the origin side of the <hyperbola>. The two negative regions lie beyond its branches. The left-hand panel shows these regions; its markers locate the <stationary points> calculated next.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1-2018-q14-signs-contours.png]
{title=Sign regions, stationary points and completed contour map of the Gaussian-weighted function}
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