2.6. Let ƒ be continuous on [0, a]. If x e [0, a], define fo(x) = f(x) and let %3D 1 fa+1(x) = - 1)"f(t) dt, n = 0, 1, 2, ... (x - a) Show that the nth derivative of f, exists and equals f.
2.6. Let ƒ be continuous on [0, a]. If x e [0, a], define fo(x) = f(x) and let %3D 1 fa+1(x) = - 1)"f(t) dt, n = 0, 1, 2, ... (x - a) Show that the nth derivative of f, exists and equals f.
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
![2.6. Let f be continuous on [0, a]. If x e [0, a], define fo(x) = f(x) and let
%3D
1
fa+1(x)
п!
n = 0, 1, 2, ..
(x -
a) Show that the nth derivative of f, exists and equals f.
b) Prove the following theorem of M. Fekete: The number of changes in sign of f
in [0, a] is not less than the number of changes in sign in the ordered set of
numbers
f(a), f1(a), . .., f,(a).
Hint. Use mathematical induction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff11bf13d-0137-4dc4-86b2-863ea249143f%2F376e86e2-8ed2-461b-a2d4-4972c83fec33%2Fwtr4ac8_processed.png&w=3840&q=75)
Transcribed Image Text:2.6. Let f be continuous on [0, a]. If x e [0, a], define fo(x) = f(x) and let
%3D
1
fa+1(x)
п!
n = 0, 1, 2, ..
(x -
a) Show that the nth derivative of f, exists and equals f.
b) Prove the following theorem of M. Fekete: The number of changes in sign of f
in [0, a] is not less than the number of changes in sign in the ordered set of
numbers
f(a), f1(a), . .., f,(a).
Hint. Use mathematical induction.
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 2 steps with 1 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,
