For a point of inflection, is a necessary condition but not a sufficient one. must also change sign before and after the point of inflection. Thus 0 is not a point of inflection because does not change sign there; is negative both slightly before and slightly after 0.

Alternatively …

when . Since is an odd-order derivative, is a point of inflection. However so we differentiate again.

when . is an even-order derivative, so is not a point of inflection.