Editing Inflection point (section)

{{short description|Point where the curvature of a curve changes sign}}
{{More footnotes|date=July 2013}}
[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]
{{Cubic graph special points.svg}}
In [[differential calculus]] and [[differential geometry]], an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (rarely '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.

For the graph of a function {{math|''f''}} of [[differentiability class]] {{math|''C''<sup>2</sup>}} (its first derivative {{math|''f'''}}, and its [[second derivative]] {{math|''f<nowiki>''</nowiki>''}}, exist and are continuous), the condition {{math|''f<nowiki>''</nowiki> {{=}} ''0}} can also be used to find an inflection point since a point of {{math|''f<nowiki>''</nowiki> {{=}}'' 0}}  must be passed to change {{math|''f<nowiki>''</nowiki>''}} from a positive value (concave upward) to a negative value (concave downward) or vice versa as {{math|''f<nowiki>''</nowiki>''}} is continuous; an inflection point of the curve is where {{math|''f<nowiki>''</nowiki> {{=}} ''0}} and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.

In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.