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==Categorization of points of inflection==

==Categorization of points of inflection==

[[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).]]

[[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).]]

  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).

  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).

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.

 

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.

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.

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。

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  x³ + 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  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 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。

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

==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.