# 拐点

此词条由Typhoid翻译

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

模板:Cubic graph special points.svg

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.

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，拐点是光滑平面曲线上的改变曲率符号的一个点。特别地，在函数图像中，拐点处函数从凹(向下)变为凸(向上) ，反之亦然。

For example, if the curve is the graph of a function *y* = *f*(*x*), of differentiability class *C*^{2}, an inflection point of the curve is where *f''*, the second derivative of f, vanishes (*f'' = 0*) and changes its sign at the point (from positive to negative or from negative to positive).^{[1]} 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, the second derivative of , vanishes (f = 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。

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.

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，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。

## Definition

定义

Inflection points in differential geometry are the points of the curve where the curvature changes its sign.^{[2]}^{[3]}

Inflection points in differential geometry are the points of the curve where the curvature changes its sign.

在微分几何中，拐点是改变曲率符号的点。

For example, the graph of the 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.

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，当其仅当一阶导数在x处具有孤立极值点时（这不同于极值点的说法），可微函数图才在(x, f(x))处拥有拐点。也就是说，在某些邻域中，该点是唯一具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上切线与曲线相交的点。

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.

下降拐点的两边导数都是负数，即在该点附近函数减小。上升拐点的两边导数都为正，即在该点附近函数增加。

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.^{[4]}

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.

对于一条代数曲线，当且仅当切线与曲线(在切点处)的交点数大于2时，非奇点为拐点。

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.

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海森曲线的交点集合一致。

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.

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.

对于由参数方程组给出的光滑曲线，若某点处曲率从正变为负或从负变为正，即改变曲率符号，则该点就是拐点。

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.

对于一条二次可微函数的光滑曲线，拐点处上的二次导数为0并且改变曲率符号。

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 ]]

从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此相反。切线是蓝色的，该处曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，并且红色的是拐点: 0,/2和]。

## A necessary but not sufficient condition

必要非充分条件

If the second derivative, *f模板:''*(*x*) exists at *x*_{0}, and *x*_{0} is an inflection point for f, then *f模板:''*(*x*_{0}) = 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*^{4}.

If the second derivative, (x)}} exists at , and is an inflection point for , then (x_{0}) 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^{4}}}.

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}}。

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模板:'*(*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*.

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.

在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域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)拐点存在的一个充分条件是:

- If
*f*(*x*) is k times continuously differentiable in a certain neighborhood of a point x with k odd and*k*≥ 3, while*f*^{(n)}(*x*_{0}) = 0 for*n*= 2, …,*k*− 1 and*f*^{(k)}(*x*_{0}) ≠ 0 then*f*(*x*) has a point of inflection at*x*_{0}.

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

如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在该处有一个拐点。

2) Another sufficient existence condition requires *f模板:''*(*x* + ε) and *f模板:''*(*x* − *ε*) to have opposite signs in the neighborhood of *x* (Bronshtein and Semendyayev 2004, p. 231).

2) Another sufficient existence condition requires (x + ε)}} and (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。

## Categorization of points of inflection

拐点的分类

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).

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)。

Points of inflection can also be categorized according to whether *f模板:'*(*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(x)}是否为0来进行分类。

- if
*f模板:'*(*x*) is zero, the point is a*stationary point of inflection*

- if
*f模板:'*(*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.

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.

驻点并不是局部极值点。普遍地，在多实变量函数的前提下，不是局部极值点的驻点被称为鞍点。

An example of a stationary point of inflection is the point (0, 0) on the graph of *y* = *x*^{3}. The tangent is the x-axis, which cuts the graph at this point.

An example of a stationary point of inflection is the point on the graph of x^{3}}}. The tangent is the -axis, which cuts the graph at this point.

一个驻点的例子是在x^{3 图上的点（0,0），其切线是x轴。
}

An example of a non-stationary point of inflection is the point (0, 0) on the graph of *y* = *x*^{3} + *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 on the graph of x^{3} + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.

一个非驻点的例子是x < sup > 3 + ax }图上的（0,0），对于任意非零的a，在原点处的切线是ax }}}}

## Functions with discontinuities

非连续性函数

Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function [math]\displaystyle{ x\mapsto \frac1x }[/math] is concave for negative x and convex for positive 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]\displaystyle{ 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也可改变凹度。他们可以通过改变垂直渐近线或非连续性来实现。例如，函数 < math > x mapsto frac1x </math > 在x为负的时候显凹性，在x为正的时候显凸性。但这个函数不具有拐点，因为0不在其定义域内。

## See also

- Hesse configuration formed by the nine inflection points of an elliptic curve海塞配置 被椭圆曲线上九个拐点所组成

- Ogee, an architectural form with an inflection point S形曲线，具有一个拐点的建筑型式曲线

- Vertex (curve), a local minimum or maximum of curvature顶点，曲线的局部最小或局部最大值点

## References

- ↑ Stewart, James (2015).
*Calculus*(8 ed.). Boston: Cengage Learning. pp. 281. ISBN 978-1-285-74062-1. - ↑
*Problems in mathematical analysis*. Baranenkov, G. S.. Moscow: Mir Publishers. 1976 [1964]. ISBN 5030009434. OCLC 21598952. - ↑ Bronshtein; Semendyayev (2004).
*Handbook of Mathematics*(4th ed.). Berlin: Springer. p. 231. ISBN 3-540-43491-7. - ↑ "Point of inflection".
*encyclopediaofmath.org*.

## Sources

Category:Differential calculus

类别: 微分

Category:Curves

类别: 曲线

Category:Analytic geometry

类别: 解析几何

This page was moved from wikipedia:en:Inflection point. Its edit history can be viewed at 拐点/edithistory