Can a function who is linear in each variable separately have a degenerate critical point that is not a saddle point?

Question

Given a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ that is linear in each variable $x_i$ separately but not in the vector $x$, e.g, $f(x)=2x_1x_4x_6-7x_1x_2x_3+15x_3x_5x_6$. This family of functions always has zero diagonal hessian, hence the hessian is not definite or semi-definite. In general, the hessian can have a zero eigenvalue and hence have a degenerate critical point. However, it sounds weird that a function that is linear in each variable separately can have a minimum or a maximum point. After all, this function is monotone in each variable. How do I classify the critical points in this case?