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Maximum and minimum. Local and global maxima and minima for cos (3π x )/ x, 0.1≤ x ≤1.1. In mathematical analysis, the maximum and minimum [a] of a function are, respectively, the largest and smallest value taken by the function. Known generically as extremum, [b] they may be defined either within a given range (the local or relative ...
Extreme value theorem. 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.
Derivative test. In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is ...
Lagrange multiplier. 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]
In mathematics, the maximum modulus principle in complex analysis states that if is a holomorphic function, then the modulus cannot exhibit a strict maximum that is strictly within the domain of . In other words, either is locally a constant function, or, for any point inside the domain of there exist other points arbitrarily close to at which ...
Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative , if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.
The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). If f : M → N is a function between metric spaces M and N, then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).
The upper envelope or pointwise maximum is defined symmetrically. For an infinite set of functions, the same notions may be defined using the infimum in place of the minimum, and the supremum in place of the maximum. For continuous functions from a given class, the lower or upper envelope is a piecewise function whose pieces are from the same ...