Given a function $f(x)$ which has up to 2 order continuous derivatives.
It is known that if $f(x)$ attains local minimum at $x=x_0$, then $f'(x_0)=0$ and $f''(x_0)\geq 0$.
I am wondering what is the necessary condition of strict local minimizer, i.e., if $f(x)$ attains strict local minimum at $x=x_0$, what can we deduce from it?
(I understand that $f'(x_0)=0$ and $f''(x_0)> 0$ is not necessary.)
Many thanks!