differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...

and

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

changes sign. In particular, in the case of the

graph of a function In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in a plane (geometry), plane and often form a P ...

, it is a point where the function changes from being

concave Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon A simple polygon that is not convex is called concave, non-convex or ...

(concave downward) to

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

(concave upward), or vice versa. For the graph of a function of

differentiability class In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

(its first derivative , and its

second derivative In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...

, exist and are continuous), the condition can also be used to find an inflection point since a point of must be passed to change from a positive value (concave upward) to a negative value (concave downward) or vice versa as is continuous; an inflection point of the curve is where 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 Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

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.

Definition

Inflection points in differential geometry are the points of the curve where the

changes its sign. For example, the graph of the

differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

has an inflection point at if and only if its

first derivative First most commonly refers to: * First, the ordinal form of the number 1 First or 1st may also refer to: Acronyms * Faint Images of the Radio Sky at Twenty-Centimeters, an astronomical survey carried out by the Very Large Array * Far Infrared a ...

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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

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. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing. For a smooth curve given by

parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...

s, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

. For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the

has an isolated zero and changes sign. In

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a non singular point of an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

is an ''inflection point'' if and only if the

intersection number In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...

of the tangent line and the curve (at the point of tangency) is greater than 2. The main motivation of this different definition, is that otherwise the set of the inflection points of a curve would not be an

algebraic set Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

. In fact, the set of the inflection points of a plane algebraic curve are exactly its non-singular points that are zeros of the Hessian determinant of its

projective completion In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the de ...

Conditions

A necessary but not sufficient condition

For a function ''f'', if its second derivative exists at and is an inflection point for , then , but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is for the function given by . In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of is the same on either side of in a

neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...

of . If this sign is positive, the point is a ''rising point of inflection''; if it is negative, the point is a ''falling point of inflection''.

Sufficient conditions

# A sufficient existence condition for a point of inflection in the case that is times continuously differentiable in a certain neighborhood of a point with odd and , is that for and . Then has a point of inflection at . # Another more general sufficient existence condition requires and to have opposite signs in the neighborhood of (

Bronshtein and Semendyayev ''Bronshtein and Semendyayev'' (often just ''Bronshtein'' or ''Bronstein'', sometimes ''BS'') (Or Handbook Of Mathematics) is the informal name of a comprehensive handbook of fundamental working knowledge of mathematics and table of formulas o ...

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 In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...

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 In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

. 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

x\mapsto \frac1x

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.

References

Sources

* * {{springer, title=Point of inflection, id=p/p073190 Differential calculus Differential geometry Analytic geometry Curves Curvature (mathematics)