Inflection Point - A Necessary But Not Sufficient Condition

A Necessary But Not Sufficient Condition

If x is an inflection point for f then the second derivative, f″(x), is equal to zero if it exists, but this condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but a undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of such a undulation point is y = x4 for x=0.

It follows from the definition that the sign of f′(x) on either side of the point (x,y) must be the same. If this is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.