mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly in

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

, a stationary point of a

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 its ...

of one variable is a point on the

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

of the function where the function's

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable

function of several real variables In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...

, a stationary point is a point on the

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

of the graph where all its

partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...

s are zero (equivalently, the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...

is zero). Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the

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

is horizontal (i.e.,

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

to the -axis). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the plane.

Turning points

A turning point is a point at which the derivative changes sign. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. If the function is twice differentiable, the stationary points that are not turning points are horizontal

inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...

s. For example, the function

x \mapsto x^3

has a stationary point at , which is also an inflection point, but is not a turning point.

Classification

Isolated stationary points of a

C^1

real valued function

f\colon \mathbb \to \mathbb

are classified into four kinds, by the

first derivative test In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abo ...

: * a local minimum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive; * a local maximum (maximal turning point or relative maximum) is one where the derivative of the function changes from positive to negative; * a rising

point of inflection In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...

(or inflexion) is one where the derivative of the function is positive on both sides of the stationary point; such a point marks a change in

concavity In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...

; * a falling point of inflection (or inflexion) is one where the derivative of the function is negative on both sides of the stationary point; such a point marks a change in concavity. The first two options are collectively known as "

local extrema In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...

". Similarly a point that is either a global (or absolute) maximum or a global (or absolute) minimum is called a global (or absolute) extremum. The last two options—stationary points that are ''not'' local extremum—are known as

saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...

s. By Fermat's theorem, global extrema must occur (for a

C^1

function) on the boundary or at stationary points.

Curve sketching

Determining the position and nature of stationary points aids in

curve sketching In geometry, curve sketching (or curve tracing) are techniques for producing a rough idea of overall shape of a plane curve given its equation, without computing the large numbers of points required for a detailed plot. It is an application of ...

of differentiable functions. Solving the equation ''f'''(''x'') = 0 returns the ''x''-coordinates of all stationary points; the ''y''-coordinates are trivially the function values at those ''x''-coordinates. The specific nature of a stationary point at ''x'' can in some cases be determined by examining the

second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...

''f''''(''x''): * If ''f''''(''x'') < 0, the stationary point at ''x'' is concave down; a maximal extremum. * If ''f''''(''x'') > 0, the stationary point at ''x'' is concave up; a minimal extremum. * If ''f''''(''x'') = 0, the nature of the stationary point must be determined by way of other means, often by noting a sign change around that point. A more straightforward way of determining the nature of a stationary point is by examining the function values between the stationary points (if the function is defined and continuous between them). A simple example of a point of inflection is the function ''f''(''x'') = ''x''³. There is a clear change of concavity about the point ''x'' = 0, and we can prove this by means of

. The second derivative of ''f'' is the everywhere-continuous 6''x'', and at ''x'' = 0, ''f''′′ = 0, and the sign changes about this point. So ''x'' = 0 is a point of inflection. More generally, the stationary points of a real valued function

f\colon \mathbb^ \to \mathbb

are those points x₀ where the derivative in every direction equals zero, or equivalently, the

is zero.

Example

For the function ''f''(''x'') = ''x''⁴ we have ''f'''(0) = 0 and ''f''''(0) = 0. Even though ''f''''(0) = 0, this point is not a point of inflection. The reason is that the sign of ''f(''x'') changes from negative to positive. For the function ''f''(''x'') = sin(''x'') we have ''f'''(0) ≠ 0 and ''f''''(0) = 0. But this is not a stationary point, rather it is a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of ''f(''x'') does not change; it stays positive. For the function ''f''(''x'') = ''x''³ we have ''f'''(0) = 0 and ''f''''(0) = 0. This is both a stationary point and a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of ''f'''(''x'') does not change; it stays positive.

References

External links

Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio
at

cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...