Extreme value

The largest and the smallest element of a set are called extreme values, absolute extrema, or extreme records.

For a differentiable function $$f$$, if $$f(x_0)$$ is an extreme value for the set of all values $$f(x)$$, and if $$x_0$$ is in the interior of the domain of $$f$$, then $$x_0$$ is a critical point, by Fermat's theorem.

Extreme values in abstract spaces with order
In the case of a general partial order one should not confuse a least element (smaller than all other) and a minimal element (nothing is smaller). Likewise, a greatest element of a poset is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A) then m = b.

Any least element or greatest element of a poset will be unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.

In a totally ordered set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element and the maximal element will also be the greatest element. Thus in a totally ordered set we can simply use the terms minimum and maximum.

If a chain is finite then it will always have a maximum and a minimum. If a chain is infinite then it need not have a maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum.

If an infinite chain S is bounded, then the closure Cl(S) of the set occasionally has a minimum and a maximum, in such case they are called the greatest lower bound and the least upper bound of the set S, respectively.

In general, if an ordered set S has a greatest element m, m is a maximal element. Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with respect to order induced by T, m is a least upper bound of S in T. The similar result holds for least element, minimal element and greatest lower bound.