= Solution
For $w=x^3e^{-x^2-y^2}$, the first <derivatives> are $w_x=x^2(3-2x^2)e^{-x^2-y^2}$ and $w_y=-2yx^3e^{-x^2-y^2}$. Both vanish at the origin. The local expansion is
$$
w(x,y)=x^3-x^5-x^3y^2+\cdots=x^3+O\!\left((x^2+y^2)^{5/2}\right).
$$
There 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.
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