This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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 de±ned 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....
View Full Document
- Fall '09
- Math, Continuous function, Suppose, real function