Editing Inflection point (section)

=== A necessary but not sufficient condition ===
For a function ''f'', if its second derivative {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}} and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|1=''f{{''}}''(''x''<sub>0</sub>) = 0}}, but this condition is not [[Sufficient condition|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 {{math|1=''x'' = 0}} for the function {{mvar|f}} given by {{math|1=''f''(''x'') = ''x''<sup>4</sup>}}.

In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|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 {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.