Maximum and minimum

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.

A real-valued function f defined on a domain X has a global (or absolute) maximum point at x∗, if f(x∗) ≥ f(x) for all x in X.

Similarly, the function has a global (or absolute) minimum point at x∗, if f(x∗) ≤ f(x) for all x in X.

Symbolically, this can be written as follows: The definition of global minimum point also proceeds similarly.

If the domain X is a metric space, then f is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) for all x in X within distance ε of x∗.

Similarly, the function has a local minimum point at x∗, if f(x∗) ≤ f(x) for all x in X within distance ε of x∗.

In both the global and local cases, the concept of a strict extremum can be defined.

For example, x∗ is a strict global maximum point if for all x in X with x ≠ x∗, we have f(x∗) > f(x), and x∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x∗ with x ≠ x∗, we have f(x∗) > f(x).

An important example is a function whose domain is a closed and bounded interval of real numbers (see the graph above).

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.

So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the greatest (or least) one.Minima For differentiable functions, Fermat's theorem states that local extrema in the interior of a domain must occur at critical points (or points where the derivative equals zero).

[5] For any function that is defined piecewise, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is greatest (or least).

feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where

For example, in the (enlargeable) figure on the right, the necessary conditions for a local maximum are similar to those of a function with only one variable.

The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure).

These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a saddle point.

For use of these conditions to solve for a maximum, the function z must also be differentiable throughout.

For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction).

This is illustrated by the function whose only critical point is at (0,0), which is a local minimum with f(0,0) = 0.

Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with (respect to order induced by T), then m is a least upper bound of S in T. Similar results hold for least element, minimal element and greatest lower bound.

The maximum and minimum function for sets are used in databases, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-decomposable aggregation functions.

Thus in a totally ordered set, we can simply use the terms minimum and maximum.

For example, the set of natural numbers has no maximum, though it has a minimum.

Local and global maxima and minima for cos(3π x )/ x , 0.1≤ x ≤1.1
The global maximum of x √ x occurs at x = e .
Peano surface , a counterexample to some criteria of local maxima of the 19th century
The global maximum is the point at the top
Counterexample: The red dot shows a local minimum that is not a global minimum
As an example, both unnormalised and normalised sinc functions above have of {0} because both attain their global maximum value of 1 at x = 0.

The unnormalised sinc function (red) has arg min of {−4.49, 4.49}, approximately, because it has 2 global minimum values of approximately −0.217 at x = ±4.49. However, the normalised sinc function (blue) has arg min of {−1.43, 1.43}, approximately, because their global minima occur at x = ±1.43, even though the minimum value is the same. [ 7 ]