C is it always true that f has an absolute maximum in

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) there. (c) Is it always true that f has an absolute maximum in (0 , 1)? If yes, prove it, if not then give a counterexample. No. E.g. f ( x ) = x is a function like this and it does not have an absolute maximum in (0 , 1). (Since it’s strictly less than 1 and it can be bigger than 1 - ε for any ε > 0.) 6. Show that the equation x = cos( x 3 ) has at least one real solution. The function f ( x ) = x - cos( x 3 ) is continuous for every x 0 (right-continuous at x = 0). We have f (0) = - 1 and f (4) = 4 - cos(64) 2 - 1 = 1. Thus f (0) < 0 , f (4) > 1 and by Bolzano’s theorem there must be an x (0 , 4) 3
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