4. Assume that a continuous function f : R –→ R is T-periodic; that is, f(x + T) = f(x) for all x E R. Prove that for every natural number n there exists Xn E [0, T] such that f(xn) = f(xn +). TT:
4. Assume that a continuous function f : R –→ R is T-periodic; that is, f(x + T) = f(x) for all x E R. Prove that for every natural number n there exists Xn E [0, T] such that f(xn) = f(xn +). TT:
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
![4. Assume that a continuous function f : R → R is T-periodic; that is, f(x +
T) = f(x) for all x E R. Prove that for every natural number n there exists
Xn E [0, T] such that f(xn) = f(xn +).
Hint: Think about the function g(x) = f(x+ ?) – f(x) and the expression
n
st0) + () +-+ (")
T.
+...+g
T(n – 1)
n
n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F189e4505-6a39-4378-a0c6-eeb6998c4c33%2Ff7s8z9_processed.png&w=3840&q=75)
Transcribed Image Text:4. Assume that a continuous function f : R → R is T-periodic; that is, f(x +
T) = f(x) for all x E R. Prove that for every natural number n there exists
Xn E [0, T] such that f(xn) = f(xn +).
Hint: Think about the function g(x) = f(x+ ?) – f(x) and the expression
n
st0) + () +-+ (")
T.
+...+g
T(n – 1)
n
n
Expert Solution

Step 1
given a continuous function is T-periodic,
defined as
to prove- for every natural number there exist
such that
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