Prove or disprove involving periodic derivatives and functions

In summary, the question is whether a function with a periodic derivative must also be periodic. The attempt at a solution suggests using trigonometric functions as a counterexample, specifically secx, cscx, or cotx. However, their periodicity is uncertain and it may be easier to disprove the argument. The hint given is to consider g(x)=f(x+p)-f(x) and its derivative g'(x)=0.
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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.
 
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  • #2
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?
 

1. What are periodic derivatives and functions?

Periodic derivatives and functions are mathematical concepts that describe the repeating pattern of a function over a certain interval. A periodic function is one that repeats itself at regular intervals, while a periodic derivative is the rate of change of that function over time.

2. How can periodic derivatives and functions be proved or disproved?

Periodic derivatives and functions can be proved by showing that the function has a repeating pattern over a certain interval and that its derivative also has a repeating pattern with the same period. It can be disproved by finding a counterexample where the function does not have a repeating pattern or the derivative does not have the same period as the function.

3. Can all functions have periodic derivatives?

No, not all functions have periodic derivatives. Only functions that have a repeating pattern over a certain interval can have periodic derivatives. Functions that do not have a repeating pattern, such as exponential or logarithmic functions, do not have periodic derivatives.

4. How can the period of a periodic function be determined?

The period of a periodic function can be determined by finding the smallest interval over which the function repeats itself. This can be done by looking at the graph of the function or by solving for the value of the variable that causes the function to repeat.

5. What is the relationship between periodic derivatives and functions?

The relationship between periodic derivatives and functions is that the derivative of a periodic function is also a periodic function with the same period. This means that the rate of change of the function over time also follows a repeating pattern. Additionally, the derivative of a periodic function can help determine the period of the function.

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