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- Summary
- I'm wondering if given that the mean of a periodic fuction is zero than the mean of all of its derivatives is zero too.

We have a periodic function ##f: \mathbb{R} \rightarrow \mathbb{R}## with period ##T, f(x+T)=f(x)##

The statement is the following: $$\frac{1}{T}\int_0^T f(x)dx =0 \implies \frac{1}{T}\int_0^T\frac{d}{dx} f(x)dx =0$$

Can you give me a hint on how to prove/disprove it? The examples I tried all confirmed this.

The statement is the following: $$\frac{1}{T}\int_0^T f(x)dx =0 \implies \frac{1}{T}\int_0^T\frac{d}{dx} f(x)dx =0$$

Can you give me a hint on how to prove/disprove it? The examples I tried all confirmed this.