= Solution
At a differentiable interior <stationary point>, all first-order changes of the scalar field vanish. In particular, the <directional derivative> along each coordinate axis vanishes, so $f_x=f_y=0$ and hence $\nabla f=\boldsymbol0$. Equivalently, the first-order differential $df=\nabla f\cdot d\boldsymbol r$ vanishes for every displacement. A differentiable interior maximum or minimum must satisfy this condition; a stationary point need not be an extremum.
The standard nondegenerate local <contour plot> models are $-x^2-y^2$ for a maximum, $x^2+y^2$ for a minimum, and $x^2-y^2$ for a <saddle point of a scalar function>. Near a maximum, nested contours have lower values farther from the centre; near a minimum they have higher values. Near a saddle, <hyperbolas> occur on opposite sides of the critical level, whose contour has two intersecting tangent directions. More general nondegenerate quadratic forms deform these circles into <ellipses>; degenerate stationary points can have more complicated contours.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2-q12-stationary-contours.png]
{title=Local contours near a maximum, a minimum and a saddle}
{description=Original quadratic models, with signed contour labels showing how the field changes away from each stationary point.}
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