In mathematics, arg max (or argmax) stands for the argument of the maximum, that is to say, the value of the given argument for which the value of the given expression attains its maximum value:

\underset{x}{\operatorname{arg\,max}} \, f(x) \quad\in\quad \{x\ |\ \forall y : f(y) \le f(x)\}

In other words,

\underset{x}{\operatorname{arg\,max}} \, f(x)

is the value of x for which f(x) has the largest value (for example, if f(x) is −|x|, then it attains its maximum value at x = 0).

This is well-defined only if the maximum is reached at a single value. Thus

x_0 = \underset{x}{\operatorname{arg\,max}} \, f(x)

holds if and only if x₀ is the unique value of x for which f(x) is maximized. So, for example,

\underset{x\in \Bbb{R}}{\operatorname{arg\,max}} (x(10-x)) = 5

since the maximum value of x(10 − x) is 25, which happens when x = 5.

However, in case the maximum is reached at many values, arg max can be extended to value a set of solutions.

Then, we have for example

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

since the maximum value of cos(x) is 1, which happens when x = 0, 2π or 4π.

arg min (or argmin) is defined analogously.

See also

• Mode (statistics)

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