Inflection points

In differential calculus, an inflection point, point of inflection, flex, or inflection (inflexion (British English)) is a point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.

A point where the curvature vanishes but does not change sign is sometimes called a point of undulation or undulation point.

In algebraic geometry an inflection point is defined slightly more generally, as a 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 are the points of the curve where the curvature changes its sign while a tangent exists.

A 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 rising point of inflection is an inflection point where the derivative has a local minimum, and a falling point of inflection is a point where the derivative has a local maximum.

For an algebraic curve, a non singular point is an inflection point if and only if the multiplicity of the intersection of the tangent line and the curve (at the point of tangency) is odd and greater than 2.

For a curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.

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