Let the function $f_{n}:\mathbb{R}\rightarrow \mathbb{R_{\ge 0}}$ be defined as $$ f_{n}(x)=\sum_{i,j=1}^{n}\frac{(-1)^{i+j}\cos(\ln \frac{i}{j})}{(ij)^{x}}\quad \forall n\in\Bbb N $$ and let also $$ F_{n}(x)=\left(\sqrt{f_{n}(x)}-\frac{1}{(n+1)^{x}}\right)^{2} \quad \forall n\in\Bbb N. $$
Is it true that for every $n$ equation $F_{n}'(x)=0$ has at most one solution in the interval $x\in(0,1)$?
My idea was to prove that difference between consecutive zeros of given derivative is $\ge 1$ Also i wrote given functions in GeoGebra program for some different $n$ and everything seems to be ok, but I don't know how to check it in general.
Regards.