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删除97字节 、 2020年12月7日 (一) 10:54
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==Categorization of points of inflection==
 
==Categorization of points of inflection==
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==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).]]
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  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).
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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)。
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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)。
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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.
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拐点也可以根据(x)}是零还是非零来分类。
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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''
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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.
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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。
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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。
 
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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.
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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。
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一个驻点的例子是在x<sup>3图上的点(0,0),其切线是x轴。
 
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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.
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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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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 }}}}
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==Functions with discontinuities==
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==Functions with discontinuities==非连续性函数
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==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.
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