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{{short description|Point where a curve crosses its tangent and the curvature changes of sign}}

{{short description|Point where the curvature of a curve changes sign}}

{{More footnotes|date=July 2013}}

{{More footnotes|date=July 2013}}

[[Image:x cubed plot.svg|thumb|150px|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]].]]

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.]]

{{Cubic graph special points.svg}}

{{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]] 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, 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.

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。

在微分和微分几何，一个拐点，点的拐点，弯曲，或屈折(英式英语: 拐点)是一个光滑的平面曲线上的一个点，曲率改变符号。特别是，在函数图像的情况下，它是一个点的功能变化从凹(向下)到凸(向上) ，或反之亦然。

If the curve is the [[graph of a function]] {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''²}}, 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'''.

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'''.

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.

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.

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.

==Definition==

==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 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 are the points of the curve where the curvature changes its sign.

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

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.

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.

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.

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.

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。

例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(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 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 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 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 [[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 [[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 multiplicity of the intersection of the tangent line and the curve (at the point of tangency) is odd and greater than 2.

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，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。

对于一条代数曲线，一个非奇点是拐点，当且仅当切线与曲线(在切点处)的交点数大于2。

F (x) sin (2x)从-/ 4到5 / 4的图; 第二[[导数是 f / 2和]]

 

[[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}}]]

 

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)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]

==A necessary but not sufficient condition==

==A necessary but not sufficient condition==

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

If the second derivative, f(x) exists at x₀, and x₀ is an inflection point for f, then f(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 x = 0 for the function f given by f(x) = 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 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。

如果二阶导数，(x)}}存在于，并且是拐点，那么(x < sub > 0 </sub >)0} ，但是这个条件对于有拐点是不充分的，即使存在任意阶的导数。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。对于 x < sup > 4 </sup > }给出的函数，波动点的一个例子是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{{'}}''(''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 {{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 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.

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，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。

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。

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:

1)拐点存在的一个充分条件是:

:If ''f''(''x'') is ''k'' times continuously differentiable in a certain neighbourhood of a point ''x'' with ''k'' odd and ''k'' ≥ 3, while ''f''^(''n'')(''x''₀)&nbsp;=&nbsp;0 for ''n'' = 2,...,''k''&nbsp;−&nbsp;1 and ''f''^(''k'')(''x''₀)&nbsp;≠&nbsp;0 then ''f''(''x'') has a point of inflection at ''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 f(x) is k times continuously differentiable in a certain neighbourhood of a point x with k odd and k ≥ 3, while f⁽ⁿ⁾(x₀)&nbsp;=&nbsp;0 for n = 2,...,k&nbsp;−&nbsp;1 and f^(k)(x₀)&nbsp;≠&nbsp;0 then f(x) has a point of inflection at x₀.

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 .

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。

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。

2) Another sufficient existence condition requires ''f′′''(''x'' + ε) and ''f′′''(''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 {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''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 f′′(x + ε) and f′′(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)。

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

==Categorization of points of inflection==

==Categorization of points of inflection==

[[Image:X to the 4th minus x.svg|250px|thumb|''y'' = ''x''⁴ – ''x'' has a 2nd derivative of zero at point (0,0), but it is not an  

[[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).]]

 

 

 

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).]]

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).

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。]

X < sup > 4 </sup >-x }在点(0,0)处有一个零的二阶导数，但它不是拐点，因为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′''(''x'') is zero or not zero.

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 f′(x) is zero or not zero.

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

拐点也可以根据 f ′(x)是否为零来分类。

拐点也可以根据(x)}是零还是非零来分类。

* if ''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 ''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''

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 {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''³}}. The tangent is the {{mvar|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 = x³. 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 x³}}. The tangent is the -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。

一个驻点拐点的例子是 x < sup > 3 </sup > }图上的点。切线是-轴，它在此处切割图形。

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 {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''³ + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''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 = x³ + 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 x³ + ax}}, for any nonzero . The tangent at the origin is the line 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。

非平稳拐点的一个例子是 x < sup > 3 </sup > + ax }图上对任意非零点的拐点。原点处的切线是直线 ax }}}} ，它在此处切割图形。

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.

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。

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 > 是凹的表示负的，凸的表示正的，但是它没有拐点，因为0不在函数的域中。