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Concept
Inflection point
Summary
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.
For the graph of a function of differentiability class C2 (f, its first derivative f, and its second derivative f'', exist and are continuous), the condition f'' = 0 can also be used to find an inflection point since a point of f'' = 0 must be passed to change f'' from a positive value (concave upward) to a negative value (concave downward) or vice versa as f'' is continuous; an inflection point of the curve is where f'' = 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.
In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.
Inflection points in differential geometry are the points of the curve where the curvature changes its sign.
For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative f' has an isolated extremum at x. (this is not the same as saying that f has an extremum). That is, in some neighborhood, x is the one and only point at which f' has a (local) minimum or maximum. If all extrema of f' are isolated, then an inflection point is a point on the graph of f at which the tangent crosses the curve.
A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing.
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