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A necessary but not sufficient condition

For a function f, if its second derivative f ″ (x) exists at x and x is an inflection point for f, then f ″ (x) = 0, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, 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 an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is x = 0 for the function f given by f (x) = x.

In the preceding assertions, it is assumed that f has some higher-order non-zero derivative at x, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of f ' (x) is the same on either side of x in a neighborhood of x. If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.