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{{More footnotes|date=July 2013}}

{{More footnotes|date=July 2013}}

[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''³}} with an inflection point at (0,0), which is also a [[stationary point]].]]

[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''³}} with an inflection point at (0,0), which is also a [[stationary point]].|链接=Special:FilePath/X_cubed_plot.svg]]

Plot of  with an inflection point at (0,0), which is also a [[stationary point.]]

Plot of  with an inflection point at (0,0), which is also a [[stationary point.]]

In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''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.

In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''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.

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the 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 (concave downward) to convex (concave upward), or vice versa.

在微分和微分几何中，拐点（英文名为inflection point，point of infection,，flex，或者inflection，英式拼写为inflextion）是光滑平面曲线上的曲率符号改变的点。在函数图像中，拐点处函数从下凹变为上凸 ，或从上凸变为下凹。

For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''²}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''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'''.

For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''²}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref name=":0">{{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'''.

For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.

For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.

例如，若曲线是可导性类的函数图像，那么在曲线拐点处二阶导数为0，并且改变了它的符号(从正到负或从负到正)。二阶导数为0但其符号不变的点有时称为波动点。

例如，若曲线{{math|1=''y'' = ''f''(''x'')}}有二阶导数，那么拐点处曲线二阶导数''f<nowiki>'''</nowiki>''为0（f<nowiki>''</nowiki>=0），并且符号改变（从正到负或从负到正）<ref name=":0" />。二阶导数为0但其符号不变的点有时称为波动点。

[[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}}]]

[[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}}|链接=Special:FilePath/Animated_illustration_of_inflection_point.gif]]

Plot of  sin(2x)}} from −/4 to 5/4; the second [[derivative is (x)  –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]

Plot of  sin(2x)}} from −/4 to 5/4; the second [[derivative is (x)  –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]

If the second derivative, (x)}} exists at , and  is an inflection point for , then (x₀)  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  0}} for the function  given by  x⁴}}.

If the second derivative, (x)}} exists at , and  is an inflection point for , then (x₀)  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  0}} for the function  given by  x⁴}}.

如果二阶导数，(x)}}在x0处存在，并且x0是该函数的拐点，那么(x < sub > 0 </sub >)0} ，那么即使存在任意阶的导数，这个条件对于有拐点也是不充分的。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。若最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点被统称为拐点。对于给定的 x < sup > 4 </sup > }的函数，波动点是0}}。

<nowiki>如果二阶导数，(x)}}在x0处存在，并且x0是该函数的拐点，那么(x < sub > 0 )0} ，那么即使存在任意阶的导数，这个条件对于有拐点也是不充分的。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。若最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点被统称为拐点。对于给定的 x < sup > 4 }的函数，波动点是0}}。</nowiki>

 

拐点的分类

拐点的分类

[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''⁴ – ''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).]]

[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''⁴ – ''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).|链接=Special:FilePath/X_to_the_4th_minus_x.svg]]

  x⁴ – 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).

  x⁴ – 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).