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第149行：
第149行： + 第154行：
第155行： − + 第162行：
第163行： − + 第174行：
第175行： − 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。+ − 第182行：
第182行： − 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。+ − 第190行：
第189行： − 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。+ + − +
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==Categorization of points of inflection==
==Categorization of points of inflection==
==拐点的分类==
[[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).]]
[[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).]]
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).
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)。
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 (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)}是零还是非零来分类。
拐点也可以根据(x)}是否为0来进行分类。
* 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''
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 on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.
An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.
一个驻点的例子是在x<sup>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 on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line 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<sup>3</sup> + 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==
==Functions with discontinuities==非连续性函数
==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>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 {{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.