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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}} 。
* 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]].
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。
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.
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。
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.
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>