Let $\Omega$ be a bounded, connected, open subset of $\Bbb R^n$ and $u$ is real analytic on $\Bbb R^n$. Let $x_0 \in \Omega$ such that $\frac{\partial u}{\partial x_i}=0$ and $\frac{\partial^2 u}{\partial x_i^2}=0, \forall i=1,\dots, n$ then can $u$ attain a local maxima at $x_0$ ?
I know that at a point of Maxima, the total derivative of the function must vanish and its Hessian is $\le 0$. But what happens if we know it is exactly equal to 0, i do not know.
Thanks for help.