Past exam of the mathematical methods II course of the University of Oxford / 2015 / 9 / e

For $w=x^3{e}^{-x^2-y^2}$, the first derivatives are $w_x=x^2(3-2x^2)e^{-x^2-y^2}$ and $w_y=-2y{x}^3{e}^{-x^2-y^2}$. Both vanish at the origin. The local expansion isThere are no constant, linear or quadratic terms, so every second derivative vanishes at the origin and the Hessian determinant is zero. The second-order test therefore cannot classify the point.

The cubic term does determine the behaviour: along $y=0$, $w(x,0)$ has the sign of $x$. Every neighbourhood contains values above and below $w(0,0)=0$, so it is neither a maximum nor a minimum. It is a nonisolated stationary point with nearby values of both signs. Indeed the whole line $x=0$ is stationary, so this is not an isolated nondegenerate quadratic saddle.