I have learnt that all local extrema occur at critical points, but not all critical points occur at local extrema. That is the rationale where we find critical points when optimization a convex function.
But consider $f(x) = x^2$ with $1\leq x\leq 2$. The minimum of $f(x)$ is 1 at $x=1$. But this local extrema is not a critical point. Is this a counterexample of the above statement?
I think the following statement is true: When $f$ is convex, a critical point is a global minimum.
