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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. [1] [2] [3] Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
By using the interior extremum theorem, the potential extrema of a function , with derivative ′, can found by solving an equation involving ′. The interior extremum theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum).
A continuous function () on the closed interval [,] showing the absolute max (red) and the absolute min (blue).. In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed and bounded interval [,], then must attain a maximum and a minimum, each at least once.
A differentiable function graph with lines tangent to the minimum and maximum. Fermat's theorem guarantees that the slope of these lines will always be zero.. In mathematics, Fermat's theorem (also known as interior extremum theorem) is a theorem which states that at the local extrema of a differentiable function, its derivative is always zero.
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.
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1]
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:
The methods given below for optimization refer to an important subclass of quasi-Newton methods, secant methods. [2] Using methods developed to find extrema in order to find zeroes is not always a good idea, as the majority of the methods used to find extrema require that the matrix that is used is symmetrical.