Consider the function $$f(x)=\prod_{n=0}^\infty\operatorname{sinc}\left(\frac{\pi \, x}{2^n}\right),\tag1$$ where $\operatorname{sinc}(z)$ denotes the sinc function. It arises as a Fourier transform of the Rvachev $\operatorname{up}(x)$ function, which is basically a shifted version of the Fabius function (see, for example $^{[1]}$$\!^{[2]}$$\!^{[3]}$). Curiously, if we take a finite partial product from $(1)$ with at least 2 terms, its Fourier transform will be a continuous piecewise-polynomial function with finite support (and with continuous derivatives of progressively higher orders as we include more terms).
We restrict our attention only to $x\ge0$. The function $f(x)$ has zeros at positive integers, and oscillates with a quickly decaying amplitude. Its signs on the intervals between consecutive zeros follow the same pattern as the Thue–Morse sequence. 
It appears that $f(x)$ has exactly one extremum on each interval between consecutive zeros (minimum or maximum, depending on its sign on that interval) — but I have not been able to find a complete rigorous proof of it. Can you propose one?
Update: I extracted the second part of my original question into a separate one and edited it significantly. The following is just an interesting observation:
Let us denote the value of the extremum on the interval $n<x<n+1$ as $\epsilon_n$. The absolute values of the extrema $|\epsilon_n|$ generally tend to decrease as $n$ increases, but they do not decrease strictly monotonically and, in fact, show quite irregular behavior, sometimes increasing sporadically. Here is how their graph looks on log scale: 
