Editing Inflection point

{{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.

==Definition==
Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|orig-year=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref>

For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]] {{mvar|f'}} has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (this is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.

A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.

For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].

For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.

In [[algebraic geometry]], a non singular point of an [[algebraic curve]] is an ''inflection point'' if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2. The main motivation of this different definition, is that otherwise the set of the inflection points of a curve would not be an [[algebraic set]]. In fact, the set of the inflection points of a plane algebraic curve are exactly its [[non-singular point]]s that are zeros of the [[Hessian determinant]] of its [[projective completion]].

[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]

==Conditions==

=== 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''.

===Sufficient conditions===
# A sufficient existence condition for a point of inflection in the case that {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|''x''<sub>0</sub>}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, is that  {{math|1=''f''{{i sup|(''n'')}}(''x''<sub>0</sub>) = 0}} for {{math|1=''n'' = 2, ..., ''k'' − 1}} and {{math|''f''{{i sup|(''k'')}}(''x''<sub>0</sub>) ≠ 0}}. Then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.
# Another more general sufficient existence condition requires {{math|''f{{''}}''(''x''<sub>0</sub> + ''ε'')}} and {{math|''f{{''}}''(''x''<sub>0</sub> − ''ε'')}} to have opposite signs in the neighborhood of&nbsp;{{math|''x''<sub>0</sub>}} ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).

==Categorization of points of inflection==
[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]

Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.
* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''
* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''

A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].

An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.

An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.

==Functions with discontinuities==
Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.

==Functions with inflection points whose second derivative does not vanish==
Some continuous functions have an inflection point even though the second derivative is never 0.  For example, the cube root function is concave upward when x is negative, and concave downward when x is positive, but has no derivatives of any order at the origin.

== See also ==
* [[Critical point (mathematics)]]
* [[Ecological threshold]]
* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]
* [[Ogee]], an architectural form with an inflection point
* [[Vertex (curve)]], a local minimum or maximum of curvature

==References==
{{reflist}}

==Sources==
* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}
* {{springer|title=Point of inflection|id=p/p073190}}

[[Category:Differential calculus]]
[[Category:Differential geometry]]
[[Category:Analytic geometry]]
[[Category:Curves]]
[[Category:Curvature (mathematics)]]