In mathematics, **arg max** stands for the **argument of the maximum**, that is to say, the set of points of the given argument for which the given function attains its maximum value:

In other words,

is the set of values of *x* for which *f*(*x*) attains its largest value *M*. For example, if *f*(*x*) is 1−|*x*|, then it attains its maximum value of 1 at *x* = 0 and only there, so .

The argmax operator is the natural complement of the **max** operator which, given the same arguments, returns the maximum value (instead of the point or points that reach that value).

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

If the maximum is reached at a single value, then one refers to the point as *the* arg max, meaning we define the arg max as a point, not a set of points. So, for example,

(rather than the singleton set ), since the maximum value of *x*(10 − *x*) is 25, which occurs for *x* = 5.

However, in case the maximum is reached at many values, arg max is a *set* of points.

Then, we have 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, the arg max is

Note also that functions do not in general attain a maximum value, and hence will in general not have an arg max: is the empty set, as *x* is unbounded on the real line. However, by the extreme value theorem (or the classical compactness argument), a continuous function on a compact interval has a maximum, and thus an arg max.

Read more about Arg Max: Arg Min

### Other articles related to "arg max":

**Arg Max**- Arg Min

... arg min stands for argument of the minimum, and is defined analogously ... For instance, are values of x for which f(x) attains its smallest value M ...

### Famous quotes containing the word max:

“I have thought of relocating, somewhere where I’d be more appreciated. California, perhaps. I could teach earthquake preparedness.”

—Wesley Strick, U.S. screenwriter, and Martin Scorsese. *Max* Cady (Robert DeNiro)