# Dictionary Definition

maximum adj : the greatest or most complete or
best possible; "maximal expansion"; "maximum pressure" [syn:
maximal] [ant: minimal, minimal]

### Noun

1 the largest possible quantity [syn: upper limit]
[ant: minimum]

3 the point on a curve where the tangent changes
from positive on the left to negative on the right [ant: minimum] [also: maxima (pl)]

# User Contributed Dictionary

see Maximum

## English

### Etymology

French from maximum.### Pronunciation

max*im-um MAAHX-im-muhm### Noun

- The highest limit.
- The greatest value of a set or other mathematical structure, esp. the global maximum or a local maximum of a function.

#### Synonyms

#### Antonyms

#### Translations

highest limit

greatest value

### Adjective

- To the highest degree.
- Use the proper dose for the maximum effect.

#### Translations

highest degree

- Finnish: maksimaalinen
- Telugu: గరిష్ఠ (garishTha)
- Spanish: máximo

### Derived terms

### Related terms

## Latin

### Etymology

neutral of maximusmaximum

# Extensive Definition

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.

maximum in Arabic: حدود عليا وحدود دنيا

maximum in German: Extremwert

maximum in French: Extremum

maximum in Icelandic: Útgildi

maximum in Dutch: Extreme waarden

maximum in Polish: Ekstremum

maximum in Portuguese: Mínimo

maximum in Finnish: Maksimi

# Synonyms, Antonyms and Related Words

abundance, acmatic, acme, affluence, all, ample sufficiency, ampleness, amplitude, apex, apical, apogee, authority, authorization, avalanche, be-all and end-all,
best, blue ribbon, bonanza, bountifulness, bountiousness, brow, bumper crop, cap, capital, ceiling, championship, chief, climactic, climax, cloud nine, command, comprehensive, consequential, considerable, consummate, control, copiousness, crest, crown, crowning, culmen, culmination, deep, directorship, dominion, edge, effectiveness, end, exhaustive, extravagance, extreme, extreme limit, extremity, exuberance, fertility, first place, first
prize, first-class, first-rate, flood, flow, foison, full, full measure, fullness, generosity, generousness, grand, grave, great, great abundance, great
plenty, greatest,
gush, head, headmost, headship, heaven, heavens, heavy, hegemony, height, high noon, highest, highest degree, highest
pitch, highest point, imperium, influence, intense, irresistible, jurisdiction, kingship, landslide, lavishness, leadership, liberality, liberalness, limit, lordship, lots, luxuriance, main, management, mastership, mastery, maximal, meridian, meridional, mighty, more than enough,
most, mountaintop, much, myriad, myriads, ne plus ultra, new
high, no place higher, noon, nth degree, numerousness, opulence, opulency, outpouring, outside, overflow, overmost, palms, paramount, paramountcy, peak, pinnacle, pitch, plenary, plenitude, plenteousness, plentifulness, plenty, point, pole, power, powerful, preeminent, presidency, prevalence, primacy, prodigality, productiveness, profuseness, profusion, quantities, record, repleteness, repletion, rich harvest, rich
vein, richness,
ridge, riot, riotousness, rule, say, scads, serious, seventh heaven,
shower, sky, sovereignty, spate, spire, stream, strong, substantiality, substantialness,
summit, summital, superabundance, superlative, supremacy, supreme, sway, teemingness, the whole,
tip, tip-top, top, top spot, top-notch, topmost, total, ultimate, upmost, upper extremity, upper
limit, uppermost,
utmost, utmost extent,
uttermost, vertex, vertical, very top, wealth, zenith, zenithal