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Unformatted text preview: L22 – Maximum and Minimum Values Def – A function f has an absolute maximum (minimum) at c if f (c) is called the of f on D. Together they are called 1 Def – A function f has a local or relative maximum (minimum) at c if there is an open interval I containing c such that for all x in I 2 2x2 If f (x) = 1 x=0 x=0 ﬁnd the absolute maximum and minimum on [ 1,1 ], ( −1, 2 ], and [ 1,2 ) . If f (x) = 1 ﬁnd the absolute extrema on [ −1, 1 ] and ( 0, 1 ] . x 3 √ x If f (x) = 2−x x≥0 −2 ≤ x < 0 ﬁnd the relative and absolute extrema. Where do they occur? Extreme Value Theorem – If f is continuous on a closed interval [ a, b ], then f has an at some c and d in [ a, b ] . 4 Fermat’s Theorem – If f has a local extremum at c, and if f (c) exists, then 5 Def – A critical number of a function f is a number c in the domain of f such that f (c) = 0 or f (c) does not exist. If f has an extreme value at c, then c is a 6 Find all critical numbers of √ f (x) = 3 x g (x) = 1 x3 − 3x 7 To ﬁnd the absolute maximum and minimum values of a continuous function f on a closed interval [ a, b ]: 1. 2. 3. 8 Find the absolute maximum and minimum of f (x) = 3(x2 + 2x) 3 on [ 4,1 ] . 1 9 Find the absolute extrema of f (x) = x 3 − 5 52 x 3 on [ 0, 8 ] . 2 10 Find the absolute extrema of f (x) = x3 − 12x − 6 on ( 1,3 ) . 11 Find the absolute maximum and minimum values of f (x) = ln(x) on [ 1, e2 ] . x 12 Find the absolute extrema of f (x) = sin2 (x) − sin(x) − 2 on [ 0, 2π ] . 13 ...
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 Fall '08
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 Calculus

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