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In mathematical analysis , the maxima and minima the respective plurals of maximum and minimum of a function , known collectively as extrema the plural of extremum , are the largest and smallest value of the function, either within a given range the local or relative extrema , or on the entire domain the global or absolute extrema. As defined in set theory , the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets , such as the set of real numbers , have no minimum or maximum. Symbolically, this can be written as follows:. A similar definition can be used when X is a topological space , since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows:.
The need to find local maxima and minima arises in many situations. The first example we will look at is very familiar, and can also be solved without using calculus. Examples of solving such problems without the use of calculus can be found in the module Quadratics. Find the dimensions of a rectangle with perimeter metres so that the area of the rectangle is a maximum. The rectangle is a square with side lengths metres. Interactive 1. The following steps provide a general procedure which you can follow to solve maxima and minima problems.
But in any case we'll be able to execute the procedure given below to find local maxima and minima without worrying over a formal definition. This procedure is just a variant of things we've already done to analyze the intervals of increase and decrease of a function , or to find absolute maxima and minima. The possibly bewildering list of possibilities really shouldn't be bewildering after you get used to them. That is, the geometric meaning of the derivative's being positive or negative is easily translated into conclusions about local maxima or minima. Notice that although the processes of finding absolute maxima and minima and local maxima and minima have a lot in common, they have essential differences.
The following theorem asserts that local extrema occur at the critical points. Theorem Suppose that f is defined on an interval I and has a local maximum or.
Далекий гул генераторов теперь превратился в громкое урчание. Чатрукьян выпрямился и посмотрел. То, что он увидел, больше напоминало вход в преисподнюю, а не в служебное помещение. Узкая лестница спускалась к платформе, за которой тоже виднелись ступеньки, и все это было окутано красным туманом. Грег Хейл, подойдя к стеклянной перегородке Третьего узла, смотрел, как Чатрукьян спускается по лестнице.
Local maximum and minimum points are quite distinctive on the graph of a function, and are therefore useful in understanding the shape of the graph.
ReplyThe terms maxima and minima refer to extreme values of a function , that is, the maximum and minimum values that the function attains.
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