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The definition of global minimum point also proceeds similarly. If the domain X is a metric space, then f is said to have a local (or relative) maximum point at the point x ∗, if there exists some ε > 0 such that f(x ∗) ≥ f(x) for all x in X within distance ε of x ∗.
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.
Assume that function f has a maximum at x 0, the reasoning being similar for a function minimum. If x 0 ∈ ( a , b ) {\displaystyle x_{0}\in (a,b)} is a local maximum then, roughly, there is a (possibly small) neighborhood of x 0 {\displaystyle x_{0}} such as the function "is increasing before" and "decreasing after" [ note 1 ] x 0 ...
Stated precisely, suppose that f is a real-valued function defined on some open interval containing the point x and suppose further that f is continuous at x. If there exists a positive number r > 0 such that f is weakly increasing on (x − r, x] and weakly decreasing on [x, x + r), then f has a local maximum at x.
In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. [1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions ...
In mathematics, the maximum-minimums identity is a relation between the maximum element of a set S of n numbers and the minima of the 2 n − 1 non-empty subsets of S.
For example, for the identity function defined on the unit interval has a global and local maximum at x = 1. It is a local maximum, since the domain of the function is the unit interval, and for any x in the unit interval that is within some distance ε (say ε = 1 for concreteness) of 1, we have f(x) < f(1). I'll update the page to take this ...
On the other hand, if X is the domain of a function f(x) and if the limit as n approaches infinity of f(x n) is L for every arbitrary sequence of points {x n} in X − x 0 which converges to x 0, then the limit of the function f(x) as x approaches x 0 is equal to L. [10] One such sequence would be {x 0 + 1/n}.