Editing Inflection point (section)

==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}}]]