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lecture22 - Lecture 22 Maximum and Minimum Values(Chapter 4...

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Lecture 22: Maximum and Minimum Values (Chapter 4, Section 22) Def. A function f has an absolute maximum at x = c if f ( x ) = 2 ¡ j x j 6 - ? f ( c ) is the of f on D . Def. f has an absolute minimum at x = c if f ( x ) = x 2 ¡ 1 6 - ? f ( c ) is the of f on D . Together they are called
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2 Def. A function f has a local (relative) maximum at c if there is an open interval I containing c such that for all x in I Def. A function f has a local (relative) minimum at c if there is an open interval I containing c such that for all x in I ex. Find all local and absolute extrema of the func- tion f ( x ) sketched below. 6 - ?
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3 Let f ( x ) = ( x 2 ¡ 2 x 1 p x ¡ 1 x > 1 . 6 - ? Find the absolute maximum and minimum values of f ( x ) on each of the following intervals: [ ¡ 2 ; 1] ( ¡ 2 ; 1) [2 ; 5] (1 ; 1 ) Find any local extrema.
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4 Extreme Value Theorem: If f is continuous on a closed interval [ a; b ], then f has at some numbers c and d in [ a; b ]. 6 - ? ex. Find the absolute extrema of f ( x ) = 1 x on [ ¡ 1 ; 1].
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