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Local maximum at x = −1− √ 15 /3, local minimum at x = −1+ √ 15 /3, global maximum at x = 2 and global minimum at x = −4. For a practical example, [ 6 ] assume a situation where someone has 200 {\displaystyle 200} feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x {\displaystyle x} is ...
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value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930.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.
It is calculated as the difference between the largest and smallest values (also known as the sample maximum and minimum). [1] It is expressed in the same units as the data. The range provides an indication of statistical dispersion. Since it only depends on two of the observations, it is most useful in representing the dispersion of small data ...
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.
The maximum of a subset of a preordered set is an element of which is greater than or equal to any other element of , and the minimum of is again defined dually. In the particular case of a partially ordered set , while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements.
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.
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 ...
For a sample set, the maximum function is non-smooth and thus non-differentiable. For optimization problems that occur in statistics it often needs to be approximated by a smooth function that is close to the maximum of the set. A smooth maximum, for example, g(x 1, x 2, …, x n) = log( exp(x 1) + exp(x 2) + … + exp(x n) )