A direct extract from my book which states.(I have attached a photo as well)
"A point P(Xo,Yo) on the curve of a function y=f(x) is a point of inflection if f"(Xo)=0 and either f"(x) changes its sign at X=Xo, or [third derivative] i.e. f"'(Xo)≠0."
We can find pretty many examples for point of inflection where f"(Xo)=0 and f"(X) changes its sign at X=Xo. Like Y=X^3 at X=0. f"(X) is negative for (-∞,0) and positive for (0, +∞). So 0 is a point of inflection.
But i am really having hard time finding an example for second condition