• Support PF! Buy your school textbooks, materials and every day products Here!

Prove or disprove involving periodic derivatives and functions

  • Thread starter tmlrlz
  • Start date
  • #1
29
0

Homework Statement


A function f(x) has a periodic derivative. In other words f ' (x + p) = f ' (x) for some real value of p. Is f(x) necessarily periodic? Prove or give a counterexample.


Homework Equations


Periodic functions and Periodic Derivatives


The Attempt at a Solution


To be honest, this question stumped me because the only functions that i can think of when the mention of periodic comes is trig functions. I'm thinking that it must be a trig function which can act as a counterexample, specifically secx,cscx or cotx. however i'm not sure if these functions are periodic in the first place and then if their derivatives are periodic. i'm quite sure their derivatives are periodic but i'm not sure if they are periodic and if they're not, that will act as a counterexample. I'm worried that this might be true, because its much easier to disprove an argument than it is to prove it. If this is true, can someone help me go about the means of proving it. Thank you.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
The derivative of f(x) can be periodic even if f(x) isn't periodic. Concentrate on finding a counterexample. There's actually another thread on this same problem. The hint given was you know if g(x)=f(x+p)-f(x), so you know g'(x)=0. Does that make g(x) zero?
 

Related Threads on Prove or disprove involving periodic derivatives and functions

  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
3
Views
1K
Replies
1
Views
1K
Replies
7
Views
562
Replies
1
Views
2K
Replies
29
Views
11K
Replies
3
Views
2K
Replies
9
Views
2K
Top