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Unformatted text preview: Finding absolute extrema for nice functions Math 8 2009, Chris Lyons Lets suppose youre given a function f : [ a,b ] R and you know that its continuous. Even this little bit of information already tells you something: Theorem 1 (Thm 3.12 in Apostol) . If g : [ , ] R is continuous, then there is at least one point [ , ] such that g ( ) is the absolute maximum of g . Similarly, there is at least one point [ , ] such that g ( ) is the absolute minimum of g . So by this theorem you know there are x-values in [ a,b ] where f takes its absolute maximum and absolute minimum... but how do you actually find these x-values? If you look at the proof of Theorem 1, its of no help: it shows you that these x-values exist, by using the completeness axiom for the real numbers, but it doesnt actually give a way of finding them. (Proofs like this are called non-constructive proofs.) But if your function f is nice enough, then you can actually find these points! Well see at the end what I mean by nice enough. First, lets make the following observation, which may look kind of useless, but isI mean by nice enough....
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