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In mathematical analysis, the maximum and minimum [a] of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum , [ b ] they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function.
The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the ...
Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. [1]
Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.
After establishing the critical points of a function, the second-derivative test uses the value of the second derivative at those points to determine whether such points are a local maximum or a local minimum. [1] If the function f is twice-differentiable at a critical point x (i.e. a point where f ′ (x) = 0), then:
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 ...
It was first proposed in 1974 by Rastrigin [1] as a 2-dimensional function and has been generalized by Rudolph. [2] The generalized version was popularized by Hoffmeister & Bäck [ 3 ] and Mühlenbein et al. [ 4 ] Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima .
A critical point (where the function is differentiable) may be either a local maximum, a local minimum or a saddle point. If the function is at least twice continuously differentiable the different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.