Editing Arg max (section)

== Examples and properties ==

For example, if <math>f(x)</math> is <math>1 - |x|,</math> then <math>f</math> attains its maximum value of <math>1</math> only at the point <math>x = 0.</math> Thus 

:<math>\underset{x}{\operatorname{arg\,max}}\, (1 - |x|) = \{ 0 \}.</math>

The <math>\operatorname{argmax}</math> operator is different from the <math>\max</math> operator. The <math>\max</math> operator, when given the same function, returns the {{em|[[Maxima and minima|maximum value]]}} of the function instead of the {{em|point or points}} that cause that function to reach that value; in other words

:<math>\max_x f(x)</math> is the element in <math>\{ f(x) ~:~ f(s) \leq f(x) \text{ for all } s \in S \}.</math>

Like <math>\operatorname{argmax},</math> max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike <math>\operatorname{argmax},</math> <math>\operatorname{max}</math> may not contain multiple elements:<ref group=note>Due to the [[Antisymmetric relation|anti-symmetry]] of <math>\,\leq,</math> a function can have at most one maximal value.</ref> for example, if <math>f(x)</math> is <math>4 x^2 - x^4,</math> then <math>\underset{x}{\operatorname{arg\,max}}\, \left( 4 x^2 - x^4 \right) = \left\{-\sqrt{2}, \sqrt{2}\right\},</math> but <math>\underset{x}{\operatorname{max}}\, \left( 4 x^2 - x^4 \right) = \{ 4 \}</math> because the function attains the same value at every element of <math>\operatorname{argmax}.</math>

Equivalently, if <math>M</math> is the maximum of <math>f,</math> then the <math>\operatorname{argmax}</math> is the [[level set]] of the maximum:

:<math>\underset{x}{\operatorname{arg\,max}} \, f(x) = \{ x ~:~ f(x) = M \} =: f^{-1}(M).</math>

We can rearrange to give the simple identity<ref group=note>This is an identity between sets, more particularly, between subsets of <math>Y.</math></ref>

:<math>f\left(\underset{x}{\operatorname{arg\,max}} \, f(x) \right) = \max_x f(x).</math>

If the maximum is reached at a single point then this point is often referred to as {{em|the}} <math>\operatorname{argmax},</math> and <math>\operatorname{argmax}</math> is considered a point, not a set of points. So, for example,

:<math>\underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, (x(10 - x)) = 5</math>

(rather than the [[Singleton (mathematics)|singleton]] set <math>\{ 5 \}</math>), since the maximum value of <math>x (10 - x)</math> is <math>25,</math> which occurs for <math>x = 5.</math><ref group="note">Note that <math>x (10 - x) = 25 - (x-5)^2 \leq 25</math> with equality if and only if <math>x - 5 = 0.</math></ref> However, in case the maximum is reached at many points, <math>\operatorname{argmax}</math> needs to be considered a {{em|set}} of points.

For example

:<math>\underset{x \in [0, 4 \pi]}{\operatorname{arg\,max}}\, \cos(x) = \{ 0, 2 \pi, 4 \pi \}</math>

because the maximum value of <math>\cos x</math> is <math>1,</math> which occurs on this interval for <math>x = 0, 2 \pi</math> or <math>4 \pi.</math>  On the whole real line

:<math>\underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \cos(x) = \left\{ 2 k \pi ~:~ k \in \mathbb{Z} \right\},</math> so an infinite set.

Functions need not in general attain a maximum value, and hence the <math>\operatorname{argmax}</math> is sometimes the [[empty set]]; for example, <math>\underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, x^3 = \varnothing,</math> since <math>x^3</math> is [[Bounded function|unbounded]] on the real line. As another example, <math>\underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \arctan(x) = \varnothing,</math> although [[Arc tangent|<math>\arctan</math>]] is bounded by <math>\pm\pi/2.</math> However, by the [[extreme value theorem]], a continuous real-valued function on a [[Interval (mathematics)|closed interval]] has a maximum, and thus a nonempty <math>\operatorname{argmax}.</math>