In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, particularly in
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
, a
stationary point is an input to a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
where the
derivativeIn calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
is zero (equivalently, the
gradientIn vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
is zero): where the function "stops" increasing or decreasing (hence the name).
For the graph of a one-dimensional function, this corresponds to a point on the graph where the
tangentIn geometry, the tangent line to a curve at a given point is the straight line that "just touches" the curve at that point...
is
parallelParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not intersect or...
to the
x-axis. For the graph of a two-dimensional function, this corresponds to a point on the graph where the tangent plane is parallel to the
xy plane.
The term is mostly used in one dimension, which this article discusses: stationary points in higher dimensions are usually referred to as
critical pointsIn calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
; see there for higher dimensional discussion.
The term "critical point" is often confused with "stationary point".
Critical pointIn calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
is more general: a critical point is
either a stationary point
or a point where the derivative is not defined.
A stationary point is always a critical point,
but a critical point is not always a stationary point: it might also be a non-differentiable point.
For a smooth function, these are interchangeable, hence the confusion; when a function is understood to be smooth, one can refer to stationary points as critical points, but when it may be non-differentiable, one should distinguish these notions.
Note that there is also a different definition of critical point in higher dimensions, where the derivative does not have full rank, but is not necessarily zero; this is not analogous to stationary points, as the function may still be changing in some direction.
Isolated stationary points of a real valued function are classified into four kinds, by the
first derivative testIn calculus, the first derivative test determines whether a given critical point of a function is a maximum, a minimum, or neither.-Introduction:...
:
- a minimal extremum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive;
- a maximal extremum (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, an inflection point, or point of inflection is a point on a curve at which the curvature changes sign. The curve change from being concave upwards to concave downwards , or vice versa...
(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 concavityIn mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex cap or upper convex.-Definition:...
- a falling point of inflection
In differential calculus, an inflection point, or point of inflection is a point on a curve at which the curvature changes sign. The curve change from being concave upwards to concave downwards , or vice versa...
(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
Notice: Global (or absolute) maxima and minima are sometimes called global (or absolute) maximal (resp.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, particularly in
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
, a
stationary point is an input to a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
where the
derivativeIn calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
is zero (equivalently, the
gradientIn vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
is zero): where the function "stops" increasing or decreasing (hence the name).
For the graph of a one-dimensional function, this corresponds to a point on the graph where the
tangentIn geometry, the tangent line to a curve at a given point is the straight line that "just touches" the curve at that point...
is
parallelParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not intersect or...
to the
x-axis. For the graph of a two-dimensional function, this corresponds to a point on the graph where the tangent plane is parallel to the
xy plane.
The term is mostly used in one dimension, which this article discusses: stationary points in higher dimensions are usually referred to as
critical pointsIn calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
; see there for higher dimensional discussion.
Versus critical point
The term "critical point" is often confused with "stationary point".
Critical pointIn calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
is more general: a critical point is
either a stationary point
or a point where the derivative is not defined.
A stationary point is always a critical point,
but a critical point is not always a stationary point: it might also be a non-differentiable point.
For a smooth function, these are interchangeable, hence the confusion; when a function is understood to be smooth, one can refer to stationary points as critical points, but when it may be non-differentiable, one should distinguish these notions.
Note that there is also a different definition of critical point in higher dimensions, where the derivative does not have full rank, but is not necessarily zero; this is not analogous to stationary points, as the function may still be changing in some direction.
Classification
Isolated stationary points of a real valued function are classified into four kinds, by the
first derivative testIn calculus, the first derivative test determines whether a given critical point of a function is a maximum, a minimum, or neither.-Introduction:...
:
- a minimal extremum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive;
- a maximal extremum (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, an inflection point, or point of inflection is a point on a curve at which the curvature changes sign. The curve change from being concave upwards to concave downwards , or vice versa...
(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 concavityIn mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex cap or upper convex.-Definition:...
- a falling point of inflection
In differential calculus, an inflection point, or point of inflection is a point on a curve at which the curvature changes sign. The curve change from being concave upwards to concave downwards , or vice versa...
(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
Notice: Global (or absolute) maxima and minima are sometimes called global (or absolute) maximal (resp. minimal) extrema. By
Fermat's theoremIn mathematics, Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. It gives a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point...
, they must occur on the boundary or at
critical pointIn calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
s, but they do not necessarily occur at stationary points.
Curve sketching
Determining the position and nature of stationary points aids in curve sketching, especially for
continuous functionIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
s. 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 derivativeIn calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...
f''(
x):
- If f''(x) < 0, the stationary point at x is a maximal extremum.
- If f''(x) > 0, the stationary point at x is 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 provided the function values exist 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. However, this is limited again in that it works only for functions that are continuous in at least a small interval surrounding the stationary point.
A simple example of a point of inflection is the function
f(
x) =
x3. There is a clear change of concavity about the point
x = 0, and we can prove this by means of
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
. 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:
Rn →
R are those
points
x0 where the derivative in every direction equals zero, or equivalently, the
gradientIn vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
is zero.
Example
At x
1 we have
f' (
x) = 0 and
f''(
x) = 0. Even though
f''(
x) = 0, this point is not a point of inflexion. The reason is that the sign of
f' (
x) changes from negative to positive.
At x
2, we have
f' (
x) 0 and
f''(
x) = 0. But, x
2 is not a stationary point, rather it is a point of inflexion. This because the concavity changes from concave upwards to concave downwards and the sign of
f' (
x) does not change; it stays positive.
At x
3 we have
f' (
x) = 0 and
f''(
x) = 0. Here, x
3 is both a stationary point and a point of inflexion. This is because the concavity changes from concave upwards to concave downwards and the sign of
f' (
x) does not change; it stays positive.
See also
- Optimization (mathematics)
In mathematics, optimization, or mathematical programming, refers to choosing the best element from some set of available alternatives.In the simplest case, this means solving problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or...
- Fermat's theorem
In mathematics, Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. It gives a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point...
- Second derivative test
In calculus, a branch of mathematics, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local maximum or a local minimum using the value of the second derivative at the point....
- Higher order derivative test
In mathematics, the higher-order derivative test is used to find maxima, minima, and points of inflexion in an nth degree polynomial's curve.-The test:Let be a differentiable function on the interval and let be a point on it such that#;...
- Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function...
- Saddle point
In mathematics, a saddle point is a point in the domain of a function of two variables which is a stationary point but not a local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
External links