Semi positive definite Hessian Matrix

Question

Given a function $f(x,y):\mathbb{R}^2\to \mathbb{R}$, $C^2$, and a point $x_{0}$ where:

1) The derivative is zero 2) The respective Hessian matrix is semi positive definite

Can I say $x_{0}$ is a local minimum point? If not, what can I say about this point and how can I reach more information about it?

Answer 1

Take $x_0=(0,0)$, $f(x,y)=x^4+y^4$, $g(x,y)=-x^4-y^4$, and $h(x,y)=x^3-y^3$. In each case:

• $f'(x_0)$ is the null function;
• the Hessian at $x_0$ is semi-definite positive (actually, it's $0$).

However, $x_0$ is a minimum of $f$, a maximum of $g$, and a saddle point of $h$. So, you have to examine how the function behaves near $x_0$ by other methods.