# Arg max

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In mathematics, the **arguments of the maxima** (abbreviated **arg max** or **argmax**) are the points, or elements, of the domain of some function at which the function values are maximized.^{[note 1]} In contrast to global maxima, which refers to the largest *outputs* of a function, arg max refers to the *inputs*, or arguments, at which the function outputs are as large as possible.

## DefinitionEdit

Given an arbitrary set *X*, a totally ordered set *Y*, and a function, , the arg max over some subset, *S*, of *X* is defined by

If *S* = *X* or *S* is clear from the context, then *S* is often left out, as in In other words, arg max is the set of points, *x,* for which *f*(*x*) attains the function's largest value (if it exists). Arg max may be the empty set, a singleton, or contain multiple elements. For example, if *f*(*x*) is 1−|*x*|, then *f* attains its maximum value of 1 only at the point *x* = 0. Thus,

- .

The *arg max* operator is different than the *max* operator. The *max* operator, when given the same function, returns the *maximum value* of the function instead of the *point or points* that cause that function to reach that value; in other words

- is the element in

Like arg max, max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike arg max, max may not contain multiple elements:^{[note 2]} for example, if *f*(*x*) is 4*x*^{2} − *x*^{4}, then , but because the function attains the same value at every element of arg max.

Equivalently, if *M* is the maximum of *f*, then the arg max is the level set of the maximum:

We can rearrange to give the simple identity^{[note 3]}

- .

If the maximum is reached at a single point then this point is often referred to as *the* arg max, and arg max is considered a point, not a set of points. So, for example,

(rather than the singleton set {5}), since the maximum value of *x*(10 − *x*) is 25, which occurs for *x* = 5.^{[note 4]} However, in case the maximum is reached at many points, arg max needs to be considered a *set* of points.

For example

since the maximum value of cos(*x*) is 1, which occurs on this interval for *x* = 0, 2π or 4π. On the whole real line

- , so an infinite set.

Functions need not in general attain a maximum value, and hence the arg max is sometimes the empty set; for example, , since is unbounded on the real line. As another example, , although *arc tan* is bounded by ±π/2. However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty arg max.

## Arg minEdit

**arg min** (or **argmin**) stands for **argument of the minimum**, and is defined analogously. For instance,

are points *x* for which *f*(*x*) attains its smallest value. It is the complementary operator of .

## See alsoEdit

## NotesEdit

**^**For clarity, we refer to the input (*x*) as*points*and the output (*y*) as*values;*compare critical point and critical value.**^**Due to the anti-symmetry of ≤, a function can have at most one maximal value.**^**This is an identity between sets, more particularly, between subsets of*Y*.**^**Note that with equality if and only if .

## ReferencesEdit

**^**"The Unnormalized Sinc Function Archived 2017-02-15 at the Wayback Machine", University of Sydney