# 更改

、 2021年9月8日 (三) 15:10

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.]]
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{{Cubic graph special points.svg}}

{{Cubic graph special points.svg}}

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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, 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''<sup>2</sup>}}, 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'''.
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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.
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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>
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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>
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.
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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.
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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.
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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.
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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.
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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>
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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.
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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]].
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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.
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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]].
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.
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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.
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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.
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If the 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|''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|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.
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If the 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|''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|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.
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If the second derivative, (x)}} exists at , and  is an inflection point for , then (x<sub>0</sub>)  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  x<sup>4</sup>}}.
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<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>
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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''.
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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.
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'''Inflection points sufficient conditions:'''

'''Inflection points sufficient conditions:'''
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Inflection points sufficient conditions:

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

1) A sufficient existence condition for a point of inflection is:
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1) A sufficient existence condition for a point of inflection is:
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1)第一充分条件：

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1)拐点存在的一个充分条件是:

: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''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.

: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''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.
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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 .
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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).
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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。
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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。
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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).

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).
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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)
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x<sup>4</sup><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>
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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.
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Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.
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* 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''
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* 若 f"(x)=0，该点为驻点拐点。

* 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''
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* 若 f"(x)≠0，该点为非驻点拐点。

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