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Unformatted text preview: g to [ a, b ]. 6. Suppose f is a real Function on ( , ). We say that x R is a fxed point For f iF f ( x ) = x . 1 2 (a) If f is diFerentiable and f ( t ) n = 1 for every real t , prove that f has at most one xed point. (b) Show that the function f dened by f ( t ) = t + 1 1 + e t has no xed point, although 0 < f ( t ) < 1 for all real t . (c) Prove that if there is a constant 0 < A < 1 such that | f ( t ) | A for all t R , then f has a xed point x . To do this, given x 1 R arbitrary, construct the sequence x n +1 = f ( x n ) n 1 , and prove that the sequence converges to some point x . Then prove that x is the xed point....
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This note was uploaded on 12/27/2011 for the course MATH 118b taught by Professor Garcia during the Fall '09 term at UCSB.
- Fall '09