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,
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.
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. The complementary operator is, of course, min.