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 r = 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 [ƒ'(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.
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 r = 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 [ƒ'(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.
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 B of this question. Thank you.
![3. In this problem we study another approach to show that certain equations have a unique solution on
f(x) for some function f
an interval [a, b]. The goal is to first write the equation in the form
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4893c46-a94c-4c81-a9fc-277ce07300a6%2Fca22f1e3-58f5-4061-9632-e07b18b4a0cc%2Fwy3a5v3_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
f(x) for some function f
an interval [a, b]. The goal is to first write the equation in the form
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
