abs_extrema

abs_extrema - Finding absolute extrema for “nice”...

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Unformatted text preview: Finding absolute extrema for “nice” functions Math 8 2009, Chris Lyons Let’s suppose you’re given a function f : [ a,b ] → R and you know that it’s continuous. Even this little bit of information already tells you something: Theorem 1 (Thm 3.12 in Apostol) . If g : [ α,β ] → R is continuous, then there is at least one point γ ∈ [ α,β ] such that g ( γ ) is the absolute maximum of g . Similarly, there is at least one point δ ∈ [ α,β ] such that g ( δ ) 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, it’s of no help: it shows you that these x-values exist, by using the completeness axiom for the real numbers, but it doesn’t 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! We’ll see at the end what I mean by “nice enough”. First, let’s make the following observation, which may look kind of useless, but isI mean by “nice enough”....
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abs_extrema - Finding absolute extrema for “nice”...

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