(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).)
(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
Related questions
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).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86c8dcbb-d46d-4c91-a740-ef32ebf33ae0%2F8e238125-ee1b-4dfb-8609-7127742e5205%2Fcjsozi_processed.png&w=3840&q=75)
Transcribed Image Text:(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
![](/static/compass_v2/shared-icons/check-mark.png)
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].
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)