Let $f:\mathbb{R}^2\to\mathbb{R}$, $f(x,y)=xye^{-(x+y)}$.
I want to check for local extrema and saddle points.
The gradient is given by:
$\nabla f(x,y)=((y-xy)e^{-(x+y)}, (x-xy)e^{-(x+y)})$
Solving this for $0$ gives the two candidates $(0,0)$ and $(1,1)$
The Hessian is given by:
$\operatorname{Hess}f(x,y)=\begin{pmatrix}e^{-(x+y)}(xy-2y)& e^{-(x+y)}(xy+1-x-y)\\e^{-(x+y)}(xy+1-x-y)&e^{-(x+y)}(xy-2x)\end{pmatrix}$
Now I investigate
$\operatorname{Hess}f(0,0)=\begin{pmatrix} 0&1\\1&0\end{pmatrix}$
$\operatorname{Hess}f(1,1)=\begin{pmatrix} -e^{-2}&0\\0&-e^{-2}\end{pmatrix}$
and see that both matrices are indefinite. For example with the Hurwitz criterion for the determinant of the minors.
Is this correct? In particular, can you spot saddle points with the Hurwitz criterion?
Thanks in advance.