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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.]]

拐点在(0,0) ，这也是一个[[驻点]]

拐点在(0,0) ，也是一个[[驻点]]。

{{Cubic graph special points.svg}}

{{Cubic graph special points.svg}}

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。

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 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 {{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'''.

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。

 

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。

 

定义

定义

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|origyear=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>

Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref name=":1">{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref name=":2">{{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.

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.

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

例如，当且仅当可微函数 {{mvar|f}} 的一阶导数 {{mvar|f'}} 在 {{mvar|x}} 处具有孤立极值时（而不是极值），函数在 {{math|(''x'', ''f''(''x''))}}处有拐点。也就是说，在某些邻域中，{{mvar|x}} 是 {{mvar|f'}} 具有（局部）最小值或最大值的唯一点。如果所有 {{mvar|f'}} 的极值都是孤立的，那么拐点就是 {{mvar|f}} 曲线图上切线穿越曲线的点。

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.

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 an [[algebraic curve]], a non singular point 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.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref>

For an [[algebraic curve]], a non singular point 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.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref>

 

The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].

The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].

The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.

其主要结果是代数曲线拐点的集合与曲线同黑塞曲线 (Hessian curve) 的交点集合一致。

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 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.

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.

必要非充分条件

必要非充分条件

If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''₀}}, and {{math|''x''₀}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''₀) {{=}} 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|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''⁴}}.

If the second derivative, {{math|''f"{{''}}''(''x'')}} exists at {{math|''x''₀}}, and {{math|''x''₀}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''₀) {{=}} 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|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''⁴}}.

 

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''.

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''.

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

前面我们假定 {{mvar|f}} 在 {{mvar|x}} 处存在高阶非零导数，但并不一定存在。如果 {{mvar|f}} 在 {{mvar|x}} 处存在高阶非零导数，第一个非零导数有奇数阶意味着f<nowiki>''</nowiki>(x)的符号在某邻域的任一边都是相同的。如果符号为正，那么这个点就是上升拐点；如果符号为负，那么这个点就是下降拐点。

'''Inflection points sufficient conditions:'''

'''Inflection points sufficient conditions:'''

拐点充分条件:

拐点充分条件:

1) A sufficient existence condition for a point of inflection is:

1) A sufficient existence condition for a point of inflection is:

1) A sufficient existence condition for a point of inflection is:

1)第一充分条件：

 

:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''^(''n'')(''x''₀)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''^(''k'')(''x''₀)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''₀}}.

:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''^(''n'')(''x''₀)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''^(''k'')(''x''₀)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''₀}}.

If  is  times continuously differentiable in a certain neighborhood of a point  with  odd and , while &nbsp;0}} for  2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and  then  has a point of inflection at .

设函数在点 {{mvar|x}} 的某邻域 {{mvar|k}} 阶连续可微，{{mvar|k}} 为奇数且 {{math|''k'' ≥ 3}}，若 {{math|''f''^(''n'')(''x''₀)&nbsp;{{=}}&nbsp;0}} ( {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}}) 且 {{math|''f''^(''k'')(''x''₀)&nbsp;≠&nbsp;0}} ，那么点 {{math|''x''₀}} 是 {{math|''f''(''x'')}} 的拐点。

2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).

2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).

2)另一个充分存在条件则要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近具有相反的符号。

2)第二充分条件：f<nowiki>''</nowiki>(x + ε) 和 f<nowiki>''</nowiki> (x&nbsp;−&nbsp;ε) 在 x 邻域符号相反。

  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).

X < sup > 4 -x }在点(0,0)处二阶导数为0，但它不是拐点，因为其四阶导数是一阶非零导数(三阶导数也是零)。

x⁴<nowiki> – x}}在点 (0,0) 处二阶导数为0，但 (0,0) 不是拐点，因为其四阶导数是一阶非零导数（三阶导数也是零）。</nowiki>

Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.

Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.

Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.

拐点也可以根据f"=(x)是否为0来进行分类。

 

拐点也可以根据(x)}是否为0来进行分类。

* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''

* 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''

* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''