Suppose that I have a real valued function of a single variable $f(x)$ which is twice differentiable in some open interval $I$.
Then, I know from calculus that if $f''(x) >0 $ on $I$, then $f$ is convex on $I$.
But, what I am wondering is that if I only know that $f''(x) > 0$ at a particular point $x_0$ in $I$, then can I say that $f(x)$ is locally convex at $x_0$? That is, can I find a small neighbourhood around $x_0$ in which $f$ is convex?
If this is true, does this idea that a positive second derivative at a point means local convexity generalize?