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第8行： − + 第22行：
第22行： + − − 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。 第33行：
第32行： − 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。+ 第68行：
第67行： + 第75行：
第75行： + 第88行：
第89行： − + − − 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]。 − + 第120行：
第119行： − 1)第一充分条件：+ 第133行：
第132行： − 2)第二充分条件：f<nowiki>''</nowiki>(x + ε) 和 f<nowiki>''</nowiki> (x − ε) 在 x 邻域符号相反。+ 第142行：
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无编辑摘要
Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point.]]
Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point.]]
{{math|1=''y'' = ''x''<sup>3</sup>}} 的拐点是(0,0) ，也是一个[[驻点]]。
{{math|1=''y'' = ''x''<sup>3</sup>}} 的函数图像，(0,0)是其拐点 ，也是[[驻点]]。
{{Cubic graph special points.svg}}
{{Cubic graph special points.svg}}
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'''.
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。
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，如切线与曲线相切的正则点至少为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。
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。
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。
对于由参数方程组给出的光滑曲线，若某点处曲率符号改变（从正变为负或从负变为正），则该点就是拐点。
对于由参数方程组给出的光滑曲线，若某点处曲率符号改变（从正变为负或从负变为正），则该点就是拐点。
[[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}}|链接=Special:FilePath/Animated_illustration_of_inflection_point.gif]]
[[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}}|链接=Special:FilePath/Animated_illustration_of_inflection_point.gif]]
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 ]]
Plot of f(x)=sin(2x) from −π/4 to 5π/4; the second derivative is f"(x)=–4sin(2x), and its sign is thus the opposite of the sign of f. Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, π/2 and π.
f(x)=sin(2x)−π/4 到 5π/4 的函数图像。该函数二阶导数为 f"(x)=–4sin(2x)，和 f 符号相反。曲线为凸时（函数在切线上方）切线颜色为蓝色，曲线为凹时（函数在切线下方）切线颜色为绿色，拐点颜色为红色：0，π/2 和 π。
==A necessary but not sufficient condition==
==A necessary but not sufficient condition==
必要非充分条件
必要非充分条件
1) A sufficient existence condition for a point of inflection is:
1) A sufficient existence condition for a point of inflection is:
1）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''<sup>(''n'')</sup>(''x''<sub>0</sub>) {{=}} 0}} for {{math|''n'' {{=}} 2, …, ''k'' − 1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>) ≠ 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>) {{=}} 0}} for {{math|''n'' {{=}} 2, …, ''k'' − 1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>) ≠ 0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.
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) 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）第二充分条件：f<nowiki>''</nowiki>(x + ε) 和 f<nowiki>''</nowiki> (x − ε) 在 x 邻域符号相反。（见《数学手册》2004版第231页）
[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''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).|链接=Special:FilePath/X_to_the_4th_minus_x.svg]]
[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''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).|链接=Special:FilePath/X_to_the_4th_minus_x.svg]]
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).
y=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><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>
y=x<sup>4</sup> – 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）。