MATH 521, Spring 2014, Assignment 7Due date: Monday, April 28Problems for submission:1. Suppose that f : [a, b] 7→ R has a continuous derivative f0on [a, b].Show that, for every  > 0, there exists a δ > 0 such that, for everyx, y ∈ [a, b], 0 < |x − y| < δ implies thatf(x) − f(y)x − y− f0(y)< .(Hint: Use the MVT and note that the found point c ∈ (x, y) satisfies0 < |c − y| < δ!)2. Chapter #5, Question #11 in Rudin. [Hint: For the second part, finda function and value x ∈ R such that f (x + h) + f(x − h) = 0 for allh > 0!]3. Suppose f : (a, b) 7→ R where (a, b) ⊂ R is an open interval and f is adifferentiable function. Assume that f0(x) 6= 0 for all x ∈ (a, b).(a) Show that f is injective on (a, b).(b) Show that there is an open interval (c, d) ⊂ R such that f [(a, b)] =(c, d) (i.e. f is surjective on (c, d)). (Note: We may have c = −∞or d = ∞.)(c) Since f bijectively maps (a, b) in (c, d) (i.e. it is injective andsurjective), we may define an inverse function g : (c, d) 7→ (a, b)where g(f(x)) = x for all x ∈ (a, b). Prove that g0(f(x)) =1f0(x)for all x ∈ (a, b).4. Consider a metric space (X, d) and a function f : S 7→ S for some com-pact set S ⊆ X. Recall from Assignment #6 that, if f is a contractionmapping, i.e. it satisfiesd(f(x), f (y)) ≤ cd(x, y)for some c ∈ [0, 1) and all x, y ∈ S, then there is a unique x∗∈ S suchthat f(x∗) = x∗.(a) Prove that if fn: S 7→ S is a contraction mapping for somen ∈ N, then f has a unique fixed point in S. [Hint: Note thatfn(f(x)) = f(fn(x)).](b) Suppose that f : [a, b] 7→ [a, b] has a continuous derivative f0on[a, b]. Prove that there is an M ∈ R such that|f(x) − f(y)| ≤ M |x − y|for all x, y ∈ [a, b]. (Hint: Use the MVT.)(c) Use part (b) to show that there is a unique x ∈ [0, 1] such thatcos(x) = x.(d) Use part (a) and (b) to show that there is a unique x ∈ [0, 1] suchthat e−x= x. [Hint: Note that f(x) itself is not a contractionmapping on [0, 1]!]Honors! Suppose f : R 7→ R is twice differentiable and has an isolatedroot at a ∈ R. Suppose furthermore that f0(a) 6= 0. Show thatthere is a δ > 0 such that the Newton’s Method iteration schemexn+1= F (xn) = xn−f(xn)f0(xn)is a contraction mapping on the interval [a − δ, a + δ] ⊂ R. (Hint:Note that F (x) is not a contraction everywhere but that the twicedifferentiability of f(x) implies f(x) and f0(x) are