Answered: Prove that if f(x) is a differentiable…

Prove that if f(x) is a differentiable function everywhere and f'(x) # 1 for any value of x, then f(x) has at most one fixed point.

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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Question

COURSE: Mathematical/Real Analsys (MVT4)

TOPIC: Mean Value Theorem

x = a is called a fixed point of a function f(x)
a = 1 are fixed points of f(x) = x², because f (0) = 0 and f(1) = 1).
f(a) :
= a (e.g. a = 0 and
Prove that if f(x) is a differentiable function everywhere and f'(x) # 1 for
value of x, then f(x) has at most one fixed point.
any
b. What does
*Hint*: Assume f(x) has two fixed points x = a and x
the mean value theorem say about the interval [a, b]?

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