Let $y(x)$ be a continuous function that is also continuous in all derivatives, and periodic in $x$ with period of length $L$, i.e. $y(x)$ and all its derivatives have the same value at $x$ as they do at $x+L$. $dy/dx$ at some $x$ between $0$ and $L$ has the value $y^\prime$. Prove that at some point or points between $0$ and $L$, $dy/dx$ has the value $-y^\prime$.
Not entirely sure that this is a true statement, but it seems to me intuitively that it has to be.
I thought this would be equivalent to saying some continuous function $f$ integrates to zero over some range $L$. So, if it has non-zero value $f_0$ somewhere, it must have value $-f_0$ somewhere else. Which also seems unavoidably true to me, but AFAICT is no easier to prove than the first statement.
So I'm stuck. Any ideas?