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  "body": "\n## Definition\n\nLet $D$ be a nonempty set, let $f \\colon D \\to \\R$ be a [[function|0XHDTE5YMZ]], and let $a \\in D$.\n\n1. The function $f$ attains its *maximum* at $a$ if $f(x) \\le f(a)$ for all $x \\in D$. Then $f(a)$ is the *maximum value* of $f$, written $\\max_{x \\in D} f(x)$.\n2. The function $f$ attains its *minimum* at $a$ if $f(x) \\ge f(a)$ for all $x \\in D$. Then $f(a)$ is the *minimum value* of $f$, written $\\min_{x \\in D} f(x)$.\n3. A point $a$ as in 1 or 2 is a *global extremum*, or *absolute extremum*, of $f$.\n\n## Remarks\n\nThe maximum value of $f$ is the [[greatest element|0Z7HDDXX3X]] of the image $f(D)$, when it exists.\nThen it equals the [[supremum|0BTMQ04556]] of $f(D)$, and the supremum is attained.\nThe maximum value is unique, but it can be attained at several points.\nFor $D \\subseteq \\R^n$, every global extremum is a [[local extremum|04CPG9K1C4]].\n\nExample: $f(x) = x^2$ on $D = [-1, 2]$ attains its maximum at $a = 2$.\nIndeed, for $x \\in [-1, 2]$ one has $\\abs{x} \\le 2$, so $x^2 \\le 4 = f(2)$.\n\nNon-example: $f(x) = x$ on $D = (0, 1)$ has no maximum.\nFor each $a \\in (0, 1)$, the point $x = (a + 1)/2$ lies in $D$ and $f(x) \u003e f(a)$.\nThe supremum of $f(D)$ is $1$, but it is not a value of $f$.\nThe [[extreme value theorem|0WS1A8E2P9]] gives a maximum and a minimum for every continuous function on a closed bounded interval.\n",
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