# extremum

extremum
/ik stree"meuhm/, n., pl. extrema /-meuh/. Math.
a maximum or minimum value of a function in a specified neighborhood.
[1900-05; < NL, n. use of neut. of L extremus EXTREME]

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in calculus, any point at which the value of a function is largest (a maximum) or smallest (a minimum). There are both absolute and relative (or local) maxima and minima. At a relative maximum the value of the function is larger than its value at immediately adjacent points, while at an absolute maximum the value of the function is larger than its value at any other point in the interval of interest. At relative maxima inside the interval, if the function is smooth rather than peaked, its rate of change, or derivative, is zero. The derivative may be zero, however, at a point where the function has neither a maximum nor a minimum, as in the case for the function x3 at x = 0. One way to determine this is by going back to the original definition and finding the value of the function at immediately adjacent points. For example, the function x3 - 3x has the derivative 3x2 - 3, which equals 0 when x is ±1. By testing nearby points, such as 0.9 and 1.1, the function is seen to have a relative minimum when x is 1 and, similarly, a relative maximum when x is -1. There is also a second-derivative test: if the derivative of a function is zero at a point, then the function will have a relative maximum or minimum if the second derivative at that point is less than or greater than 0, respectively, the test failing if it equals 0. Relative maxima can also occur at points at which the derivative fails to exist, and these points must also be tested.

The theory of extrema applies to practical problems of optimization, such as finding the dimensions for a container that will hold the maximum volume for a given amount of material used in its construction. Locating the extreme points also aids in graphing functions.

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Universalium. 2010.

### См. также в других словарях:

• extremum — [ ɛkstremɔm ] n. m. • 1929; mot lat., d apr. maximum ♦ Math. Maximum ou minimum de la valeur d une fonction. Extremum absolu sur un intervalle, lorsque la fonction n admet aucun maximum supérieur, ou aucun minimum inférieur. Des extremums. ●… …   Encyclopédie Universelle

• extremum — extrémum s. n. Trimis de siveco, 10.08.2004. Sursa: Dicţionar ortografic ﻿ EXTRÉMUM s.m. (mat.) Extrem. [< fr. extrémum, cf. lat. extremum]. Trimis de LauraGellner, 13.09.2007. Sursa: DN …   Dicționar Român

• extremum — n. 1. the point located farthest from the middle of something. Syn: extreme point, extreme, extremity. [WordNet 1.5] 2. the most extreme possible amount or value. Syn: peak. [WordNet 1.5] …   The Collaborative International Dictionary of English

• extremum — [ek strē′məm] n. pl. extrema [ek strēmə] [ModL < L, an end, neut. of extremus: see EXTREME] Math. the maximum or minimum value of a function …   English World dictionary

• Extremum — « Maximum » et « Minimum » redirigent ici. Pour les autres significations, voir Maximum (homonymie) et Minimum (homonymie).  Les deux pluriels du substantif « maximum » étant « maximums » et « max …   Wikipédia en Français

• Extremum — Maximum; Spitze; Höchstwert; Hochpunkt; Spitzenwert; Maximalwert; Tiefpunkt; Nadir; Minimum * * * Extremum   das, s/...ma …   Universal-Lexikon

• Extremum — In der Mathematik ist ein Extremwert (oder Extremum; Plural: Extrema) der Überbegriff für lokales und globales Maximum und Minimum. Ein lokales Maximum ist der Wert der Funktion an einer Stelle, in deren Umgebung die Funktion keine größeren Werte …   Deutsch Wikipedia

• Extremum — ekstremumas statusas T sritis fizika atitikmenys: angl. extremum vok. Extremum, n rus. экстремум, m pranc. extrémum, m …   Fizikos terminų žodynas

• extremum — ekstremumas statusas T sritis fizika atitikmenys: angl. extremum vok. Extremum, n rus. экстремум, m pranc. extrémum, m …   Fizikos terminų žodynas

• extrémum — ekstremumas statusas T sritis fizika atitikmenys: angl. extremum vok. Extremum, n rus. экстремум, m pranc. extrémum, m …   Fizikos terminų žodynas

• Extremum — Ex|tre|mum das; s, ...ma <aus lat. extremum »das Äußerste«; vgl. ↑extrem> svw. ↑Extremwert …   Das große Fremdwörterbuch

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