Solution
= Solution
At the origin the <Hessian matrix> is
$$
H_g(0,0)=\begin{pmatrix}2&0\\0&-2\end{pmatrix},\qquad\det H_g(0,0)=-4<0.
$$
Thus the <second-derivative test> gives a \b[<saddle point>]. Its opposite signs can also be checked without a test formula: $g(x,0)=x^2e^{-x^2/2}>0$ for $x\ne0$, whereas $g(0,y)=-y^2e^{-y^2/2}<0$ for $y\ne0$. Both types of values occur arbitrarily close to the zero value at the origin, excluding either a <local maximum> or a <local minimum>.