Prove this property of the Hessian

Question

I have been reading about the hessian for a scholar work about optimization and I find this property:

Let be $H_{P_0}$ the determinant of the hessian matrix for the Lagrangian function $\mathscr{L}(x,y,\lambda)$ evaluated in the critical point $P_0$, then:

• if $H_{P_0}< 0$, $P_0$ in a minimum
• if $H_{P_0}>0 , P_0$ is a maximum

But I didn´t found the prove and I was wondering what happens when the determinat is cero?

Answer 1

In the 1D case, the second derivative test is inconclusive when the second derivative vanishes at the critical point. You can see this by looking at $f(x)=x^3,g(x)=x^4,x_0=0$; the second derivative test gives the same information but the behavior is different. The analogous thing happens in higher dimensions, including in constrained problems.

tl;dr: You can't get any information in this case.