Search results
Results from the WOW.Com Content Network
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value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930.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.
The behavior of solutions beyond this local interval can vary depending on the properties of f and the domain over which f is defined. For instance, if f is globally Lipschitz, then the local interval of existence of each solution can be extended to the entire real line and all the solutions are defined over the entire R.
Finding global maxima and minima is the goal of mathematical 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 ...
For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. [1] In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem.
The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them.
Similarly, if the supremum of belongs to , it is a maximum or greatest element of . For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element.
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.
In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form , = (,), (,) = ((,)),or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value ...