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 $x\geq y$, 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 $U\geq s$ for all elements $s\in S$. A lower bound for S is an element L of X such that $L\leq s$ for all elements $s\in S$. 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 $x\vee y$ and $x\wedge y$ respectively, satisfying the following properties:

• Idempotence: $x\vee x=x=x\wedge x;\,$
• Commutativity: $x\vee y=y\vee x;~x\wedge y=y\wedge x;\,$
• Associativity: $x\vee (y\vee z)=(x\vee y)\vee z;~x\wedge (y\wedge z)=(x\wedge y)\wedge z;\,$
• Absorption: $x\wedge (x\vee y)=x;~x\vee (x\wedge y)=x.\,$

These are characterising properties of a lattice.

Critical points

For a differentiable function f, if f(x₀) is an extreme value for the set of all values f(x), and if f(x₀) is in the interior of the domain of f, then x₀ is a critical point.

See also

• Minima and maxima