(e) Using parts (a) and (b), show that the equation tan(3r) = √2r+1 has a unique solution on (0, 1). Hint: Formulate this equation as z=f(z) for some function f(x). Then use the results from part (a) and part (b).
(e) Using parts (a) and (b), show that the equation tan(3r) = √2r+1 has a unique solution on (0, 1). Hint: Formulate this equation as z=f(z) for some function f(x). Then use the results from part (a) and part (b).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer Part C of this question. Thank you.
![3. In this problem we study another approach to show that certain equations have a unique solution on
an interval [a, b]. The goal is to first write the equation in the form x = f(x) for some function f
defined on [a, b].
(a) Suppose that f is continuous on [a, b] and that a < f(x) < b for all x € [a, b]. Prove that there
exists c = (a, b) such that c= f(c). Hint: Consider the function g(x) = x-f(x).
(b) Suppose, in addition, that f is differentiable on (a, b) and that f'(x)| < 1 for all x € (a, b). Prove
that there is a unique point c € (a, b) such that c= f(c). Hint: Use MVT.
(c) Using parts (a) and (b), show that the equation tan(3r) = √2x+1 has a unique solution on
(0, 1).
Hint: Formulate this equation as x = f(x) for some function f(x). Then use the results from part
(a) and part (b).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4893c46-a94c-4c81-a9fc-277ce07300a6%2Ff2ffd7b1-89be-41de-9d8e-705f92b1e0d0%2F4rf6tl15_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. In this problem we study another approach to show that certain equations have a unique solution on
an interval [a, b]. The goal is to first write the equation in the form x = f(x) for some function f
defined on [a, b].
(a) Suppose that f is continuous on [a, b] and that a < f(x) < b for all x € [a, b]. Prove that there
exists c = (a, b) such that c= f(c). Hint: Consider the function g(x) = x-f(x).
(b) Suppose, in addition, that f is differentiable on (a, b) and that f'(x)| < 1 for all x € (a, b). Prove
that there is a unique point c € (a, b) such that c= f(c). Hint: Use MVT.
(c) Using parts (a) and (b), show that the equation tan(3r) = √2x+1 has a unique solution on
(0, 1).
Hint: Formulate this equation as x = f(x) for some function f(x). Then use the results from part
(a) and part (b).
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