= Solution
Near a nondegenerate <stationary point>, the quadratic part of the <Taylor expansion> determines the local <contour plot>. For a strict <local maximum>, the model $f=f_0-X^2-Y^2$ gives nested closed <level sets>, with values decreasing as one moves outward. For a strict <local minimum>, the model $f=f_0+X^2+Y^2$ gives the same nested shape with values increasing outward. Unequal nonzero quadratic coefficients stretch the <circles> into <ellipses> after a rotation to the <Hessian matrix>'s <eigenvectors>.
For a nondegenerate <saddle point of a scalar function>, the model $f=f_0+X^2-Y^2$ gives <hyperbolas>. Levels above $f_0$ open along the positive-curvature direction; levels below $f_0$ open along the negative-curvature direction. The critical <level set> is the two crossing lines $Y=\pm X$. Degenerate <stationary points> can have different local geometry, so these are the standard nondegenerate sketches.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2002/paper-2-q12-local-contours.png]
{title=Local contours at a maximum, a minimum, and a saddle point}
{description=Original quadratic-model sketches with contour values labelled relative to the central value zero. Maxima and minima have closed contours with opposite value ordering; the saddle has crossing zero contours and hyperbolic nearby levels.}
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