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If the second derivative, (x)}} exists at , and  is an inflection point for , then (x₀)  0}}, but this condition is not 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  0}} for the function  given by  x⁴}}.

If the second derivative, (x)}} exists at , and  is an inflection point for , then (x₀)  0}}, but this condition is not 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  0}} for the function  given by  x⁴}}.

如果二阶导数，(x)}}存在于，并且是拐点，那么(x < sub > 0 </sub >)0} ，但是这个条件对于有拐点是不充分的，即使存在任意阶的导数。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。如果最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点通常都被称为拐点。对于 x < sup > 4 </sup > }给出的函数，波动点的一个例子是0}}。

如果二阶导数，(x)}}在x0处存在，并且x0是该函数的拐点，那么(x < sub > 0 </sub >)0} ，那么即使存在任意阶的导数，这个条件对于有拐点也是不充分的。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。若最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点被统称为拐点。对于给定的 x < sup > 4 </sup > }的函数，波动点是0}}。

If  is  times continuously differentiable in a certain neighborhood of a point  with  odd and , while &nbsp;0}} for  2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and  then  has a point of inflection at .

If  is  times continuously differentiable in a certain neighborhood of a point  with  odd and , while &nbsp;0}} for  2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and  then  has a point of inflection at .

如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在。

如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在该处有一个拐点。

2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).

2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).

2)另一个充分存在条件要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近有相反的符号。

2)另一个充分存在条件则要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近具有相反的符号。

== See also ==

== See also ==

* [[Critical point (mathematics)]]

* [[Critical point (mathematics)]]临界点

* [[Ecological threshold]]

* [[Ecological threshold]]生态阈值

* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]

* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]海塞配置  被椭圆曲线上九个拐点所组成

* [[Ogee]], an architectural form with an inflection point

* [[Ogee]], an architectural form with an inflection point S形曲线，具有一个拐点的建筑型式曲线

* [[Vertex (curve)]], a local minimum or maximum of curvature

* [[Vertex (curve)]], a local minimum or maximum of curvature顶点，曲线的局部最小或局部最大值点