Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
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.
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
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS