In order to find the nature of critical points of this function
$$f(x, y, z) = xy + xz + yz -x +z$$
Solving de system of equations given by $\nabla f = 0$ I end up with the point $(-1, 0, 1)$. If the function has a maximum, minimum or settle point, it will be at this point.
For the Hessian matrix I found the determinant is $0$:
$$\det(H_f) = \begin{vmatrix} 0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0 \end{vmatrix} = 0$$
So I conclude nothing on the nature of that critical point. Where do I go from here? What techniques could I usually go for when I'm caught up in this inconclusive test?
Thank you in advance!
The determinant is $2$, not $0$. Actually, the eigenvalues are $2$ (once) and $-1$ (twice). In particular, the point is a saddle point.