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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.

For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.

例如，如果曲线是可导性类的函数图像，那么曲线的一个拐点就是 f < nowiki > </nowiki > 的二阶导数，消失了(f < nowiki > </nowiki > = 0)并且改变了它的符号(从正到负或从负到正)。二阶导数消失但其符号不变的点有时称为波动点或波动点。

例如，若曲线是可导性类的函数图像，那么在曲线拐点处二阶导数为0，并且改变了它的符号(从正到负或从负到正)。二阶导数为0但其符号不变的点有时称为波动点。

For example, the graph of the differentiable function has an inflection point at  if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that  has an extremum).  That is, in some neighborhood,  is the one and only point at which  has a (local) minimum or maximum.  If all extrema of  are isolated, then an inflection point is a point on the graph of  at which the tangent crosses the curve.

For example, the graph of the differentiable function has an inflection point at  if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that  has an extremum).  That is, in some neighborhood,  is the one and only point at which  has a (local) minimum or maximum.  If all extrema of  are isolated, then an inflection point is a point on the graph of  at which the tangent crosses the curve.

例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上的一个点，在这个点上切线与曲线相交。

例如，当其仅当一阶导数在x处具有孤立极值点时（这不同于极值点的说法），可微函数图才在(x, f(x))处拥有拐点。也就是说，在某些邻域中，该点是唯一具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上切线与曲线相交的点。

Plot of  sin(2x)}} from −/4 to 5/4; the second [[derivative is (x)  –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]

Plot of  sin(2x)}} from −/4 to 5/4; the second [[derivative is (x)  –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]

从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]

从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此相反。切线是蓝色的，该处曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，并且红色的是拐点: 0,/2和]。

 

An example of a stationary point of inflection is the point  on the graph of  x³}}. 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  x³}}. The tangent is the -axis, which cuts the graph at this point.

一个驻点的例子是在x<sup>3图上的点（0,0），其切线是x轴。

一个驻点的例子是在x<sup>3 图上的点（0,0），其切线是x轴。