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Moved page from wikipedia:en:Inflection point (history)
此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
{{short description|Point where a curve crosses its tangent and the curvature changes of sign}}
{{More footnotes|date=July 2013}}
[[Image:x cubed plot.svg|thumb|150px|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} 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]]
{{Cubic graph special points.svg}}
In [[differential calculus]], an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion'''{{cn|reason=I did calculus in a British school and we spelled it "CT" not "X".|date=June 2019}}) is a point on a [[Continuous function|continuous]] [[plane curve]] at which the curve changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.
In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuous plane curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.
在微分中，拐点、拐点、弯曲点或拐点(英式英语: 拐点)是连续平面曲线上的一个点，在这个点上，曲线从凹(向下凹)变为凸(向上凹) ，反之亦然。
If the curve is the [[graph of a function]] {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, this means that the [[second derivative]] of {{mvar|f}} vanishes and changes sign at the point. A point where the second derivative vanishes but does not change sign is sometimes called a '''point of undulation''' or '''undulation point'''.
If the curve is the graph of a function , of differentiability class , this means that the second derivative of vanishes and changes sign at the point. A point where the second derivative vanishes but does not change sign is sometimes called a point of undulation or undulation point.
如果曲线是可微性类的函数图像，这意味着二阶导数消失并改变点的符号。二阶导数消失但不改变符号的点有时称为波动点或波动点。
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.
In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to 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.
在代数几何中，拐点的定义更为广泛一些，它是指切线与曲线相交处的一个正则点，从而使得曲线至少有3个点，而波动点或超高点则是指切线与曲线相交处的一个点，从而使得曲线至少有4个点。
==Definition==
Inflection points 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 are the points of the curve where the curvature changes its sign.
拐点是曲率改变其符号的曲线上的点。
A [[differentiable function]] has an inflection point at (''x'', ''f''(''x'')) if and only if its [[derivative|first derivative]], ''f′'', has an [[isolated point|isolated]] [[extremum]] at ''x''. (This is not the same as saying that ''f'' has an extremum). That is, in some neighborhood, ''x'' is the one and only point at which ''f′'' has a (local) minimum or maximum. If all [[extremum|extrema]] of ''f′'' are [[isolated point|isolated]], then an inflection point is a point on the graph of ''f'' at which the [[tangent]] crosses the curve.
A differentiable function has an inflection point at (x, f(x)) if and only if its first derivative, f′, has an isolated extremum at x. (This is not the same as saying that f has an extremum). That is, in some neighborhood, x is the one and only point at which f′ has a (local) minimum or maximum. If all extrema of f′ are isolated, then an inflection point is a point on the graph of f at which the tangent crosses the curve.
一个可微函数在(x，f (x)处有一个拐点当且仅当它的一阶导数 f ′在 x 处有一个孤立极值。(这不等于说 f 有极值)。也就是说，在某些邻域中，x 是 f ′有(局部)最小值或最大值的唯一点。如果 f ′的所有极值都是孤立的，那么拐点就是 f 图上的一个点，在这个点上切线与曲线相交。
A ''falling point of inflection'' is an inflection point where the derivative has a local minimum, and a ''rising point of inflection'' is a point where the derivative has a local maximum.
A falling point of inflection is an inflection point where the derivative has a local minimum, and a rising point of inflection is a point where the derivative has a local maximum.
拐点的下降点是拐点，其中导数有一个局部极小值，拐点的上升点是导数有一个局部极大值的点。
For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[Multiplicity (mathematics)|multiplicity]] of the intersection of the tangent line and the curve (at the point of tangency) is odd and greater than 2.<ref>{{cite web|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 multiplicity of the intersection of the tangent line and the curve (at the point of tangency) is odd and greater than 2.
对于一条代数曲线，一个非奇点是一个拐点，当且仅当切线与曲线(在切点处)相交的重数为奇数且大于2。
For a 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 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.
对于由参数方程组给出的曲线，如果一个点的符号曲率从正变为负或从负变为正，也就是说，改变符号，那么这个点就是拐点。
For 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 twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.
对于两次可微函数，一个拐点是图中的一个点，在这个点上二阶导数有一个孤立的零和变号。
[[Image:Animated illustration of inflection point.gif|500px|thumb|Plot of ''f''(''x'') = sin(2''x'') from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is ''f{{''}}''(''x'') = –4sin(2''x''), and its sign is thus the opposite of the sign of ''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}}]]
Plot of f(x) = sin(2x) from −/4 to 5/4; the second [[derivative is f/2 and ]]
F (x) sin (2x)从-/ 4到5 / 4的图; 第二[[导数是 f / 2和]]
==A necessary but not sufficient condition==
If the second derivative, ''f{{'}}{{'}}''{{space|hair}}(''x'') exists at ''x''<sub>0</sub>, and ''x''<sub>0</sub> is an inflection point for ''f'', then ''f{{'}}{{'}}''{{space|hair}}(''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 ''x'' = 0 for the function ''f'' given by ''f''(''x'') = ''x''<sup>4</sup>.
If the second derivative, f(x) exists at x<sub>0</sub>, and x<sub>0</sub> is an inflection point for f, then f(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 x = 0 for the function f given by f(x) = x<sup>4</sup>.
如果二阶导数 f (x)存在于 x 子0 / 子，x 子0 / 子是 f 的拐点，那么 f (x 子0 / 子)0，但是这个条件对于有拐点是不充分的，即使存在任何阶的导数。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)。如果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，拐点和起伏点通常都被称为拐点。一个波动点的例子是函数 f 的 x0，f (x) xsup 4 / sup 给出。
In the preceding assertions, it is assumed that ''f'' has some higher-order non-zero derivative at ''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 ''f{{'}}''(''x'') is the same on either side of ''x'' in a [[neighborhood (mathematics)|neighborhood]] of ''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 f has some higher-order non-zero derivative at 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 f(x) is the same on either side of x in a neighborhood of x. 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.
在前面的断言中，我们假定 f 在 x 处有一些高阶非零导数，但这并不一定是这种情况。如果是这样的话，第一个非零导数为奇数阶的条件意味着 f (x)的符号在 x 的一个邻域中 x 的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。
'''Inflection points sufficient conditions:'''
Inflection points sufficient conditions:
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)拐点存在的充分条件是:
:If ''f''(''x'') is ''k'' times continuously differentiable in a certain neighbourhood of a point ''x'' with ''k'' odd and ''k'' ≥ 3, while ''f''<sup>(''n'')</sup>(''x''<sub>0</sub>) = 0 for ''n'' = 2,...,''k'' − 1 and ''f''<sup>(''k'')</sup>(''x''<sub>0</sub>) ≠ 0 then ''f''(''x'') has a point of inflection at ''x''<sub>0</sub>.
If f(x) is k times continuously differentiable in a certain neighbourhood of a point x with k odd and k ≥ 3, while f<sup>(n)</sup>(x<sub>0</sub>) = 0 for n = 2,...,k − 1 and f<sup>(k)</sup>(x<sub>0</sub>) ≠ 0 then f(x) has a point of inflection at x<sub>0</sub>.
如果 f (x)是 k 次连续可微的，而 f sup (n) / sup (xsub 0 / sub)0是 n 2，... ，k-1和 f sup (k) / sup (xsub 0 / sub)≠0，则 f (x)在 x 子0 / sub 处有拐点。
2) Another sufficient existence condition requires ''f′′''(''x'' + ε) and ''f′′''(''x'' − ''ε'') to have opposite signs in the neighborhood of ''x'' ([[Bronshtein and Semendyayev]] 2004, p. 231).
2) Another sufficient existence condition requires f′′(x + ε) and f′′(x − ε) to have opposite signs in the neighborhood of x (Bronshtein and Semendyayev 2004, p. 231).
2)另一个充分存在条件要求 f ′(x +)和 f ′(x -)在 x 的邻域上有相反的符号(Bronshtein 和 Semendyayev，2004，p. 231)。
==Categorization of points of inflection==
[[Image:X to the 4th minus x.svg|250px|thumb|''y'' = ''x''<sup>4</sup> – ''x'' has a 2nd derivative of zero at point (0,0), but it is not an
[[Image:X to the 4th minus x.svg|250px|thumb|y = x<sup>4</sup> – x has a 2nd derivative of zero at point (0,0), but it is not an
[图片: x ^ 4减去 x.svg | 250px | thumb | y x sup 4 / sup-x 在点(0,0)处有一个0的二阶导数，但它不是一个
inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]
inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]
因为四阶导数是一阶高阶非零导数(三阶导数也是零) ，所以四阶拐点是零。]
Points of inflection can also be categorized according to whether ''f′''(''x'') is zero or not zero.
Points of inflection can also be categorized according to whether f′(x) is zero or not zero.
拐点也可以根据 f ′(x)是否为零来分类。
* if ''f′''(''x'') is zero, the point is a ''[[stationary point]] of inflection''
* if ''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]].
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.
一个驻点的曲折变化并不是局部极值。更广泛地说，在多个实变量函数的背景下，一个不是局部极值的驻点被称为鞍点。
An example of a stationary point of inflection is the point (0,0) on the graph of ''y'' = ''x''<sup>3</sup>. The tangent is the ''x''-axis, which cuts the graph at this point.
An example of a stationary point of inflection is the point (0,0) on the graph of y = x<sup>3</sup>. The tangent is the x-axis, which cuts the graph at this point.
驻点的一个例子是 y x sup 3 / sup 图上的点(0,0)。切线是 x 轴，它在这一点上切割图形。
An example of a non-stationary point of inflection is the point (0,0) on the graph of ''y'' = ''x''<sup>3</sup> + ''ax'', for any nonzero ''a''. The tangent at the origin is the line ''y'' = ''ax'', which cuts the graph at this point.
An example of a non-stationary point of inflection is the point (0,0) on the graph of y = x<sup>3</sup> + ax, for any nonzero a. The tangent at the origin is the line y = ax, which cuts the graph at this point.
非平稳拐点的一个例子是 yxsup 3 / sup + ax 图上的点(0,0) ，对于任意非零的 a。原点处的切线是直线 yax，它与图形在这一点相交。
==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.
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 and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.
有些函数在没有拐点的情况下改变凹度。相反，它们可以改变垂直渐近线或不连续性周围的凹度。例如，函数 math x mapsto frac1x / math 凹表示负，凸表示正，但是它没有拐点，因为0不在函数的域中。
== 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 calculus
类别: 微分
[[Category:Curves]]
Category:Curves
类别: 曲线
[[Category:Analytic geometry]]
Category:Analytic geometry
类别: 解析几何
<noinclude>
<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude>
[[Category:待整理页面]]
{{short description|Point where a curve crosses its tangent and the curvature changes of sign}}
{{More footnotes|date=July 2013}}
[[Image:x cubed plot.svg|thumb|150px|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} 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]]
{{Cubic graph special points.svg}}
In [[differential calculus]], an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion'''{{cn|reason=I did calculus in a British school and we spelled it "CT" not "X".|date=June 2019}}) is a point on a [[Continuous function|continuous]] [[plane curve]] at which the curve changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.
In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuous plane curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.
在微分中，拐点、拐点、弯曲点或拐点(英式英语: 拐点)是连续平面曲线上的一个点，在这个点上，曲线从凹(向下凹)变为凸(向上凹) ，反之亦然。
If the curve is the [[graph of a function]] {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, this means that the [[second derivative]] of {{mvar|f}} vanishes and changes sign at the point. A point where the second derivative vanishes but does not change sign is sometimes called a '''point of undulation''' or '''undulation point'''.
If the curve is the graph of a function , of differentiability class , this means that the second derivative of vanishes and changes sign at the point. A point where the second derivative vanishes but does not change sign is sometimes called a point of undulation or undulation point.
如果曲线是可微性类的函数图像，这意味着二阶导数消失并改变点的符号。二阶导数消失但不改变符号的点有时称为波动点或波动点。
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.
In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to 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.
在代数几何中，拐点的定义更为广泛一些，它是指切线与曲线相交处的一个正则点，从而使得曲线至少有3个点，而波动点或超高点则是指切线与曲线相交处的一个点，从而使得曲线至少有4个点。
==Definition==
Inflection points 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 are the points of the curve where the curvature changes its sign.
拐点是曲率改变其符号的曲线上的点。
A [[differentiable function]] has an inflection point at (''x'', ''f''(''x'')) if and only if its [[derivative|first derivative]], ''f′'', has an [[isolated point|isolated]] [[extremum]] at ''x''. (This is not the same as saying that ''f'' has an extremum). That is, in some neighborhood, ''x'' is the one and only point at which ''f′'' has a (local) minimum or maximum. If all [[extremum|extrema]] of ''f′'' are [[isolated point|isolated]], then an inflection point is a point on the graph of ''f'' at which the [[tangent]] crosses the curve.
A differentiable function has an inflection point at (x, f(x)) if and only if its first derivative, f′, has an isolated extremum at x. (This is not the same as saying that f has an extremum). That is, in some neighborhood, x is the one and only point at which f′ has a (local) minimum or maximum. If all extrema of f′ are isolated, then an inflection point is a point on the graph of f at which the tangent crosses the curve.
一个可微函数在(x，f (x)处有一个拐点当且仅当它的一阶导数 f ′在 x 处有一个孤立极值。(这不等于说 f 有极值)。也就是说，在某些邻域中，x 是 f ′有(局部)最小值或最大值的唯一点。如果 f ′的所有极值都是孤立的，那么拐点就是 f 图上的一个点，在这个点上切线与曲线相交。
A ''falling point of inflection'' is an inflection point where the derivative has a local minimum, and a ''rising point of inflection'' is a point where the derivative has a local maximum.
A falling point of inflection is an inflection point where the derivative has a local minimum, and a rising point of inflection is a point where the derivative has a local maximum.
拐点的下降点是拐点，其中导数有一个局部极小值，拐点的上升点是导数有一个局部极大值的点。
For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[Multiplicity (mathematics)|multiplicity]] of the intersection of the tangent line and the curve (at the point of tangency) is odd and greater than 2.<ref>{{cite web|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 multiplicity of the intersection of the tangent line and the curve (at the point of tangency) is odd and greater than 2.
对于一条代数曲线，一个非奇点是一个拐点，当且仅当切线与曲线(在切点处)相交的重数为奇数且大于2。
For a 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 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.
对于由参数方程组给出的曲线，如果一个点的符号曲率从正变为负或从负变为正，也就是说，改变符号，那么这个点就是拐点。
For 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 twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.
对于两次可微函数，一个拐点是图中的一个点，在这个点上二阶导数有一个孤立的零和变号。
[[Image:Animated illustration of inflection point.gif|500px|thumb|Plot of ''f''(''x'') = sin(2''x'') from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is ''f{{''}}''(''x'') = –4sin(2''x''), and its sign is thus the opposite of the sign of ''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}}]]
Plot of f(x) = sin(2x) from −/4 to 5/4; the second [[derivative is f/2 and ]]
F (x) sin (2x)从-/ 4到5 / 4的图; 第二[[导数是 f / 2和]]
==A necessary but not sufficient condition==
If the second derivative, ''f{{'}}{{'}}''{{space|hair}}(''x'') exists at ''x''<sub>0</sub>, and ''x''<sub>0</sub> is an inflection point for ''f'', then ''f{{'}}{{'}}''{{space|hair}}(''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 ''x'' = 0 for the function ''f'' given by ''f''(''x'') = ''x''<sup>4</sup>.
If the second derivative, f(x) exists at x<sub>0</sub>, and x<sub>0</sub> is an inflection point for f, then f(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 x = 0 for the function f given by f(x) = x<sup>4</sup>.
如果二阶导数 f (x)存在于 x 子0 / 子，x 子0 / 子是 f 的拐点，那么 f (x 子0 / 子)0，但是这个条件对于有拐点是不充分的，即使存在任何阶的导数。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)。如果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，拐点和起伏点通常都被称为拐点。一个波动点的例子是函数 f 的 x0，f (x) xsup 4 / sup 给出。
In the preceding assertions, it is assumed that ''f'' has some higher-order non-zero derivative at ''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 ''f{{'}}''(''x'') is the same on either side of ''x'' in a [[neighborhood (mathematics)|neighborhood]] of ''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 f has some higher-order non-zero derivative at 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 f(x) is the same on either side of x in a neighborhood of x. 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.
在前面的断言中，我们假定 f 在 x 处有一些高阶非零导数，但这并不一定是这种情况。如果是这样的话，第一个非零导数为奇数阶的条件意味着 f (x)的符号在 x 的一个邻域中 x 的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。
'''Inflection points sufficient conditions:'''
Inflection points sufficient conditions:
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)拐点存在的充分条件是:
:If ''f''(''x'') is ''k'' times continuously differentiable in a certain neighbourhood of a point ''x'' with ''k'' odd and ''k'' ≥ 3, while ''f''<sup>(''n'')</sup>(''x''<sub>0</sub>) = 0 for ''n'' = 2,...,''k'' − 1 and ''f''<sup>(''k'')</sup>(''x''<sub>0</sub>) ≠ 0 then ''f''(''x'') has a point of inflection at ''x''<sub>0</sub>.
If f(x) is k times continuously differentiable in a certain neighbourhood of a point x with k odd and k ≥ 3, while f<sup>(n)</sup>(x<sub>0</sub>) = 0 for n = 2,...,k − 1 and f<sup>(k)</sup>(x<sub>0</sub>) ≠ 0 then f(x) has a point of inflection at x<sub>0</sub>.
如果 f (x)是 k 次连续可微的，而 f sup (n) / sup (xsub 0 / sub)0是 n 2，... ，k-1和 f sup (k) / sup (xsub 0 / sub)≠0，则 f (x)在 x 子0 / sub 处有拐点。
2) Another sufficient existence condition requires ''f′′''(''x'' + ε) and ''f′′''(''x'' − ''ε'') to have opposite signs in the neighborhood of ''x'' ([[Bronshtein and Semendyayev]] 2004, p. 231).
2) Another sufficient existence condition requires f′′(x + ε) and f′′(x − ε) to have opposite signs in the neighborhood of x (Bronshtein and Semendyayev 2004, p. 231).
2)另一个充分存在条件要求 f ′(x +)和 f ′(x -)在 x 的邻域上有相反的符号(Bronshtein 和 Semendyayev，2004，p. 231)。
==Categorization of points of inflection==
[[Image:X to the 4th minus x.svg|250px|thumb|''y'' = ''x''<sup>4</sup> – ''x'' has a 2nd derivative of zero at point (0,0), but it is not an
[[Image:X to the 4th minus x.svg|250px|thumb|y = x<sup>4</sup> – x has a 2nd derivative of zero at point (0,0), but it is not an
[图片: x ^ 4减去 x.svg | 250px | thumb | y x sup 4 / sup-x 在点(0,0)处有一个0的二阶导数，但它不是一个
inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]
inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]
因为四阶导数是一阶高阶非零导数(三阶导数也是零) ，所以四阶拐点是零。]
Points of inflection can also be categorized according to whether ''f′''(''x'') is zero or not zero.
Points of inflection can also be categorized according to whether f′(x) is zero or not zero.
拐点也可以根据 f ′(x)是否为零来分类。
* if ''f′''(''x'') is zero, the point is a ''[[stationary point]] of inflection''
* if ''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]].
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.
一个驻点的曲折变化并不是局部极值。更广泛地说，在多个实变量函数的背景下，一个不是局部极值的驻点被称为鞍点。
An example of a stationary point of inflection is the point (0,0) on the graph of ''y'' = ''x''<sup>3</sup>. The tangent is the ''x''-axis, which cuts the graph at this point.
An example of a stationary point of inflection is the point (0,0) on the graph of y = x<sup>3</sup>. The tangent is the x-axis, which cuts the graph at this point.
驻点的一个例子是 y x sup 3 / sup 图上的点(0,0)。切线是 x 轴，它在这一点上切割图形。
An example of a non-stationary point of inflection is the point (0,0) on the graph of ''y'' = ''x''<sup>3</sup> + ''ax'', for any nonzero ''a''. The tangent at the origin is the line ''y'' = ''ax'', which cuts the graph at this point.
An example of a non-stationary point of inflection is the point (0,0) on the graph of y = x<sup>3</sup> + ax, for any nonzero a. The tangent at the origin is the line y = ax, which cuts the graph at this point.
非平稳拐点的一个例子是 yxsup 3 / sup + ax 图上的点(0,0) ，对于任意非零的 a。原点处的切线是直线 yax，它与图形在这一点相交。
==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.
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 and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.
有些函数在没有拐点的情况下改变凹度。相反，它们可以改变垂直渐近线或不连续性周围的凹度。例如，函数 math x mapsto frac1x / math 凹表示负，凸表示正，但是它没有拐点，因为0不在函数的域中。
== 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 calculus
类别: 微分
[[Category:Curves]]
Category:Curves
类别: 曲线
[[Category:Analytic geometry]]
Category:Analytic geometry
类别: 解析几何
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