# Extreme value

"Minimum" and "Maximum" redirect here. For minimum or maximum of a function, see maxima and minima.

The "largest" and the "smallest" element of a set are called **extreme values**. In general we need to distinguish various possible meanings of "largest" and "smallest"

## Linear order

In a linearly ordered set, such as the real numbers, any two elements *x* and *y* are comparable. We suppose for definiteness that , and so we may define the **maximum** of the set {*x*,*y*} to be *x* and the **minimum** to be *y*. By iteration we may define the maximum and minimum of any (non-empty) finite set.

Let *S* be a subset of a linearly ordered set (*X*,<). An *upper bound* for *S* is an element *U* of *X* such that for all elements . A *lower bound* for *S* is an element *L* of *X* such that for all elements . A set is *bounded* if it has both lower and upper bounds. In general a set need not have either an upper or a lower bound.

A *supremum* for *S* is an upper bound which is less than or equal to any other upper bound for *S*; an *infimum* is a lower bound for *S* which is greater than or equal to any other lower bound for *S*. In general a set with upper bounds need not have a supremum; a set with lower bounds need not have an infimum. The supremum or infimum of *S*, if one exists, is unique.

A *maximum* for *S* is an upper bound which is in *S*; a *minimum* for *S* is a lower bound which is in *S*. A maximum is necessarily a supremum, but a supremum for a set need not be a maximum (that is, need not be an element of the set); similarly an infimum need not be a minimum. As noted above, any finite set has a maximum and minimum which are thus its supremum and infimum.

The fundamental axiom for the real numbers is that every non-empty bounded set has a supremum and an infimum.

### Algebraic properties

Maximum and minimum are binary operations on a linearly ordered set, sometimes written and respectively, satisfying the following properties:

These are characterising properties of a lattice.

## Critical points

For a differentiable function *f*, if *f*(*x*_{0}) is an extreme value for the set of all values *f*(*x*), and if *f*(*x*_{0}) is in the interior of the domain of *f*, then *x*_{0} is a critical point.