(3) Suppose that ƒ : [0, 1] → R is a real function. with ƒ([0, 1]) ≤ [0, 1]. Show that thereexists an a € [0, 1] with f(a) = a. (Hint: Consider the function g(x) = x − f(x).)

Answered: (3) Suppose that f: [0, 1] → R is a…

(3) Suppose that f: [0, 1] → R is a real function. with f([0, 1]) ≤ [0, 1]. Show that there exists an a € [0, 1] with f(a) = a. (Hint: Consider the function g(x) = x − f(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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Question

(3) Suppose that ƒ : [0, 1] → R is a real function. with ƒ([0, 1]) ≤ [0, 1]. Show that there
exists an a € [0, 1] with f(a) = a. (Hint: Consider the function g(x) = x − f(x).)

Expert Solution

Step 1

Sol:-

Let g(x) = x - f(x). Then g(x) is also a function from [0, 1] to R.

Notice that

g(0) = 0 - f(0) = -f(0)

and

g(1) = 1 - f(1) = 1 - f(1),

and

g([0, 1]) = [0 - f(0), 1 - f(1)] = [-f(0), 1 - f(1)] ⊆ [0, 1].

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