 Local and global maxima and minima for cos(3π x)/ x, 0.1≤ x≤1.1 In mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighbourhood (local extremum) or on the function domain in its entirety (global extremum). Image File history File links Extrema_example. ...
Image File history File links Extrema_example. ...
For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In mathematics, the domain of a function is the set of all input values to the function. ...
Definitions
A real-valued function f' defined on the real line is said to have a local maximum point at the point x∗, if there exists some ε > 0, such that f(x∗) ≥ f(x) when |x − x∗| < ε. The value of the function at this point is called maximum of the function. Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In mathematics, the real line is simply the set of real numbers. ...
On a graph of a function, its local maxima will look like the tops of hills.
Similarly, a function has a local minimum point at x∗, if f(x∗) ≤ f(x) when |x − x∗| < ε. The value of the function at this point is called minimum of the function.
On a graph of a function, its local minima will look like the bottoms of valleys.
A function has a global maximum point at x∗, if f(x∗) ≥ f(x) for all x.
Similarly, a function has a global minimum point at x∗, if f(x∗) ≤ f(x) for all x.
Any global maximum (minimum) point is also a local maximum (minimum) point; however, a local maximum or minimum point need not also be a global maximum or minimum point.
Terminology: The terms local and global are synonymous with relative and absolute respectively. Also extremum is an inclusive term that includes both maximum and minimum: a local extremum is a local or relative maximum or minimum, and a global extremum is a global or absolute maximum or minimum.
Restricted domains: There may be maxima and minima for a function whose domain does not include all real numbers. A real-valued function, whose domain is any set, can have a global maximum and minimum. There may also be local maxima and local minima points, but only at points of the domain set where the concept of neighborhood is defined. A neighborhood plays the role of the set of x such that |x − x∗| < ε. In mathematics, the domain of a function is the set of all input values to the function. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
A continuous (real-valued) function on a compact set always takes maximum and minimum values on that set. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above). The neighborhood requirement precludes a local maximum or minimum at an endpoint of an interval. However, an endpoint may still be a global maximum or minimum. Thus it is not always true, for finite domains, that a global maximum (minimum) must also be a local maximum (minimum). In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
Terminology: The term optimum can replace either one of the terms maximum or minimum, depending on the context. Some optimization problems (see next paragraph) search for a global maximum value while others search for a global minimum value.
Finding maxima and minima Finding global maxima and minima is the goal of optimization. If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one. In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ...
A continuous function in a closed interval has a minimum (blue) and a maximum (red). ...
Local extrema can be found by Fermat's theorem, which states that they must occur at critical points. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test. Fermats theorem is a theorem in real analysis, named after Pierre de Fermat. ...
In mathematics, a critical point (or critical number) is a point on the domain of a function where: one dimension: the derivative is equal to zero or does not exist: it is points that are either stationary points or non-differentiable points. ...
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. ...
For any function that is defined piecewise, one finds maxima (or minima) by finding the maximum (or minimum) of each piece separately; and then seeing which one is biggest (or smallest). In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ...
Examples - The function x2 has a unique global minimum at x = 0.
- The function x3 has no global or local minima or maxima. Although the first derivative (3x2) is 0 at x = 0, this is an inflection point.
- The function x3/3 − x has first derivative x2 − 1 and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative we can see that −1 is a local maximum and +1 is a local minimum. Note that this function has no global maximum or minimum.
- The function |x| has a global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.
- The function cos(x) has infinitely many global maxima at 0, ±2π, ±4π, …, and infinitely many global minima at ±π, ±3π, ….
- The function 2 cos(x) − x has infinitely many local maxima and minima, but no global maximum or minimum.
- The function cos(3πx)/x with 0.1 ≤ x ≤ 1.1 has a global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. (See figure at top of page.)
- The function x3 + 3x2 − 2x + 1 defined over the closed interval (segment) [−4,2] has two extrema: one local maximum at x = −1−√15⁄3, one local minimum at x = −1+√15⁄3, a global maximum at x = 2 and a global minimum at x = −4. (See figure at right)
Image File history File links Extrema. ...
Image File history File links Extrema. ...
Plot of y = x3 with inflection point of (0,0). ...
Functions of more variables For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure at the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum because of the possibility of a saddle point. For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum. Image File history File links No higher resolution available. ...
Image File history File links No higher resolution available. ...
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. ...
Plot of y = x3 with a saddle-point at (0,0). ...
In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
In mathematics, the second partial derivatives test is a method in multivariable calculus used to determine if a critical point (x, y) is a minimum, maximum or saddle point. ...
See also Wikipedia does not have an article with this exact name. ...
Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ...
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 limit inferior and limit superior (also called infimum limit and supremum limit) of a sequence can be thought of as limiting bounds on the sequence. ...
A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ...
The largest and the smallest element of a set are called extreme values, or extreme records. ...
External links - Thomas Simpson's work on Maxima and Minima at Convergence
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