Consider a differentiable function f : R → R and assume that supx∈R |f′(x)| < 1. Show that there is x ∈ R such thatf(x) = x. Hint: Show that the sequence so = 0, $1 = f(so),..., Sn+1=f(sn) converges.

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Answered: Consider a differentiable function f :…

Consider a differentiable function f : R → R and assume that supx∈R |f′(x)| < 1. Show that there is x ∈ R such that f(x) = x.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider a differentiable function f : R → R and assume that sup_x∈R |f′(x)| < 1. Show that there is x ∈ R such that

f(x) = x.

Hint: Show that the sequence so = 0, $1 = f(so),..., Sn+1
=
f(sn) converges.

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