# 更改

、 2021年9月8日 (星期三)

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>}}.
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* 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。

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]].
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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.
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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.
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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.
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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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