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Encyclopedia > Stationary point
Stationary points (red pluses) and inflection points (green circles). The stationary points in this graph are all relative maxima or relative minima.
Stationary points (red pluses) and inflection points (green circles). The stationary points in this graph are all relative maxima or relative minima.

In mathematics, particularly in calculus, a stationary point is an input to a function where the gradient is zero. For the graph of a one-dimensional function, this corresponds to a point on the graph where the tangent is parallel 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. Image File history File links Graph shows stationary pts (red pluses) and inflection pts (green circles). ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... Partial plot of a function f. ... Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... Parallel 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. ...

Saddle points (coincident stationary points and inflection points). Here one is rising and one is a falling inflection point.
Saddle points (coincident stationary points and inflection points). Here one is rising and one is a falling inflection point.

An inflection point is a point where the concavity changes. A point of inflection is not necessarily a stationary point. All inflection points have the property of f''(x) = 0 but the reverse is not necessarily true. Image File history File links Graph showing coincident stationary pts and inflection pts. ... Plot of y = x3 with inflection point of (0,0). ... It has been suggested that Convex function be merged into this article or section. ...


Stationary points of a real valued function f: RR are classified into four kinds:

See also: maxima and minima
  • 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 (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
  • 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

Notice: Global (or absolute) maxima and minima are sometimes called global (or absolute) maximal (resp. minimal) extrema. While they may occur at stationary points, they are not actually an example of a stationary point. A graph illustrating local min/max and global min/max points In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the function takes the largest (maximum), or smallest (minimum) value either within a given neighbourhood (local extrema), or on the... It has been suggested that Convex function be merged into this article or section. ...


Determining the position and nature of stationary points aids in curve sketching, especially for continuous 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 f''(x): In mathematics, the graph of a function is the collection of all pairs (x, f(x)) of the function. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

  • 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 calculus. The second derivative of f is the everywhere-continuous 6x, and at x = 0, f′′ = 0, and the sign changes about this point. So x = 0 is a point of inflection. Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ...


More generally, the stationary points of a real valued function f: RnR are those points x0 where the derivative in every direction equals zero, or equivalently, the gradient is zero. Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ...


Example

At x1 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 x2, we have f' (x) ne 0 and f''(x) = 0. But, x2 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 x3 we have f' (x) = 0 and f''(x) = 0. Here, x3 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

In calculus, a branch of mathematics, the first derivative test determines whether a given critical point of a function is a maximum, a minimum, or neither. ... In calculus, a branch of mathematics, the second derivative test determines whether a given stationary point of a function (where its first derivative is zero) is a maximum, a minimum, or neither. ... In mathematics, the higher order derivative test is used to find maxima, minima and points of inflection in an nth degree polynomials curve. ... Fermats theorem is a theorem in real analysis, named after Pierre de Fermat. ... In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...

External link

  • Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio at cut-the-knot

  Results from FactBites:
 
NationMaster - Encyclopedia: Stationary point (1286 words)
The stationary points in this graph are all relative maxima or relative minima.
A point of inflection is not necessarily a stationary point.
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero (equivalently, the gradient is zero): where the function "stops" increasing or decreasing (hence the name).
Spartanburg SC | GoUpstate.com | Spartanburg Herald-Journal (810 words)
The term is mostly used in one dimension, which this article discusses: stationary points in higher dimensions are usually referred to as critical points; see there for higher dimensional discussion.
The term "critical point" is often confused with "stationary point".
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.
  More results at FactBites »

 
 

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