更改

跳到导航 跳到搜索
删除220字节 、 2021年9月8日 (星期三)
无编辑摘要
第98行: 第98行:  
If the second derivative, {{math|''f"{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 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''<sup>4</sup>}}.
 
If the second derivative, {{math|''f"{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 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''<sup>4</sup>}}.
   −
如果二阶导数 f"(x) 在点 {{math|''x''<sub>0</sub>}} 处存在,且 {{math|''x''<sub>0</sub>}} 是该函数的拐点,那么f"(x0)=0。然而,即使存在任意阶的导数,这也只是拐点的必要非充分条件。在这种情况下,若最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等),则该点是拐点;若最低阶非零导数为偶数阶,则该点不是拐点,而是波动点。但在代数几何中,波动点也是拐点。例如函数 {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}} 的一个波动点是 {{math|''x'' {{=}} 0}} 。
+
如果二阶导数 f"(x) 在点 {{math|''x''<sub>0</sub>}} 处存在,且 {{math|''x''<sub>0</sub>}} 是该函数的拐点,那么f"(x0)=0。然而,即使存在任意阶的导数,这也只是拐点的必要非充分条件。在这种情况下,若最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等),则该点是拐点;若最低阶非零导数为偶数阶,则该点不是拐点,而是波动点。但在代数几何中,波动点也是拐点。例如,函数 {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}} 的波动点是 {{math|''x'' {{=}} 0}} 。
      第154行: 第154行:  
* 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''
 
* 若 f"(x)=0,该点为驻点拐点。
 
* 若 f"(x)=0,该点为驻点拐点。
   
* if {{math|''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''
 
* 若 f"(x)≠0,该点为非驻点拐点。
 
* 若 f"(x)≠0,该点为非驻点拐点。
第162行: 第161行:  
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#Mathematical discussion|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#Mathematical discussion|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.
+
驻点拐点不是局部极值点。在多实变量函数中,不是局部极值点的驻点通常被称为鞍点。
 
  −
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 {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. 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 {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. 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  on the graph of  x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.
+
驻点拐点例子:{{math|(0, 0)}} 为函数 {{math|''y'' {{=}} ''x''<sup>3</sup>}} 的驻点 。切线为 {{mvar|x}} 轴,在{{math|(0, 0)}}与函数相切。
 
  −
一个驻点的例子是在x<sup>3 图上的点(0,0),其切线是x轴。
        第178行: 第172行:  
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 }}}}
+
非驻点拐点例子:{{math|(0, 0)}}为函数 {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}} 的非驻点({{mvar|a}}<nowiki>为任意非零常数),在原点处的切线是ax }}}}</nowiki>
     
29

个编辑

导航菜单