Maximum (mathematics)

in mathematics is the maximum the largest value from a given quantity. The term occurs in the following, easily different meanings:

• as maximum function, whose function value is the largest of their arguments, for example [itex] \ max (- 5.3) =3< /math>, [itex] \ max (5,4,3) =5< /math>. The arguments participate real numbers or general elements of a totally arranged quantity.
• as name for the largest element of any (also infinite) subset of one totally or partially arranged quantity. The largest element of a quantity does not have to give it, for example has {1, 2, 3,…} no maximum. Thatlargest element should not be confounded with the least upper bound. This is thus not necessary the smallest upper barrier, belonged for the quantity, by which the maximum is to be determined.
• as extreme value of a function, thus as the largest value, which a function in a certainRange assumes. Between local and global maxima one differentiates.

examples

[itex] \ max (\ {\ frac {1} {n} \ avoided n \ in \ mathbb {N} \}) = 1< /math>

[itex] \ max (\ {- \ frac {1} {n} \ avoided n \ in \ mathbb {N} \})< /math> existed not

< math> \ sup (\ {- \ frac {1} {n} \ avoided n \ in \ mathbb {N} \}) = 0< /math>

To real numbers of A, b applies: [itex] \ max (A, b) = \ frac {a+b} {2} + \ frac {|A-B|} {2}< /math>