# 更改

、 2020年12月7日 (一) 11:58

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.
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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.
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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 ]]
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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.

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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