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 (its first derivative to have opposite signs in the neighborhood of (Bronshtein and Semendyayev 2004, p. 231).
Categorization of points of inflection
Points of inflection can also be categorized according to whether is zero or nonzero. if is zero, the point is a stationary point of inflection if is not zero, the point is a non-stationary point of inflection
A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.
An example of a stationary point of inflection is the point on the graph of . The tangent is the -axis, which cuts the graph at this point.
An example of a non-stationary point of inflection is the point on the graph of , for any nonzero . The tangent at the origin is the line , which cuts the graph at this point.
Functions with discontinuities Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.
Functions with inflection points whose second derivative does not vanish Some continuous functions have an inflection point even though the second derivative is never 0. For example, the cube root function is concave upward when x is negative, and concave downward when x is positive, but has no derivatives of any order at the origin.
See also Critical point (mathematics) Ecological threshold Hesse configuration formed by the nine inflection points of an elliptic curve Ogee, an architectural form with an inflection point Vertex (curve), a local minimum or maximum of curvature
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