Inflection points and the second derivative

Wikipedia describes an inflection point as follows:

In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.

Since we assume the function to be smooth enough we can use the following characterisation of convex/concave: A function $f$ is locally convex at $x$ if $f''(x)> 0$ and locally concave at $x$ if $f''(x)< 0$. So your comment is correct: If the second derivative changes its sign at $x_0,$ it is an inflection point but not vice versa as the counterexample in the comment proves.

EDIT: I changed the inequalities to strict inequalities.