Etymology
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maximum (n.)

"the greatest amount, quantity, or degree," 1740, from French maximum and directly from Latin maximum (plural maxima), neuter of maximus "greatest," which is superlative of magnus "great, large, big" (of size), "abundant" (of quantity), "great, considerable" (of value), "strong, powerful" (of force); of persons, "elder, aged," also, figuratively, "great, mighty, grand, important," from PIE *mag-samo-, superlative form of root *meg- "great."

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maximum (adj.)

"greatest, at the maximum," 1834, from maximum (n.).

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maxi- 

word-forming element meaning "maximum, very large or very long for its kind," abstracted from maximum.

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maximal (adj.)

"of the highest or maximum value," 1872, from Latin maximus "greatest" (see maximum (n.)) + -al (1). Related: Maximally.

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maximize (v.)

"to make as great as possible, raise or increase to the highest degree," 1802, formed in English from maximum + -ize; first attested in Bentham, who used it often. Related: Maximized; maximizing.

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*meg- 
Proto-Indo-European root meaning "great."

It forms all or part of: acromegaly; Almagest; Charlemagne; maestro; magisterial; magistral; magistrate; Magna Carta; magnate; magnitude; magnum; magnanimity; magnanimous; magni-; Magnificat; magnificence; magnificent; magnify; magniloquence; magniloquent; Magnus; maharajah; maharishi; mahatma; Mahayana; Maia; majesty; major; major-domo; majority; majuscule; master; maxim; maximum; may (v.2) "to take part in May Day festivities;" May; mayor; mega-; megalo-; mickle; Mister; mistral; mistress; much; omega.

It is the hypothetical source of/evidence for its existence is provided by: Armenian mets "great;" Sanskrit mahat- "great, mazah- "greatness;" Avestan mazant- "great;" Hittite mekkish "great, large;" Greek megas "great, large;" Latin magnus "great, large, much, abundant," major "greater," maximus "greatest;" Middle Irish mag, maignech "great, large;" Middle Welsh meith "long, great."
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Betamax (n.)
1975, proprietary name (Sony), from Japanese beta-beta "all over" + max, from English maximum.
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off-peak (adj.)

"that is not at the maximum," 1906, originally in reference to electrical systems, from off- (adj.) (see off (prep.)) + peak (n.).

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max (v.)

"to reach the maximum level," by 1986, colloquial, from maximize or related words. Related: Maxed; maxing. As a noun, by 1811 in reference to a kind of gin said to be the best, apparently an abbreviation of French maxime.

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dragster (n.)

"hot rod or constructed car designed for maximum engine efficiency with no regard for style," 1954, from drag (n.) in the racing sense + -ster, perhaps abstracted from roadster.

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