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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.
The Turán graph T(n,r) is an example of an extremal graph. It has the maximum possible number of edges for a graph on n vertices without (r + 1)-cliques. This is T(13,4). Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence ...
Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine the extrema.
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 graph of f is a concave up parabola, ... which can be a difficult task. The usual numerical algorithms are much more efficient for finding local extrema, ...
The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are inflection points.. In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero.
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]
[e] The extremum [] is called a local maximum if everywhere in an arbitrarily small neighborhood of , and a local minimum if there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema , depending on whether the first derivatives of the continuous functions are respectively ...