Find all points of inflection for the function f (x) = x3.
We want to find where the second derivative changes sign, so first we need to find the second derivative. The first derivative is
f '(x) = 3x2
and the second derivative is
f "(x) = 6x.
The second derivative is never undefined, and the only root of the second derivative is x = 0. The sign of f " does change there, since f"(x) is negative for x < 0 and positive for x > 0.
Therefore x = 0 is an inflection point. Looking at the graph of f (x), this makes sense: