Assume that $f(x)$ is positive and decreasing on $[0,1]$ with $f(0)=1$ and $f(1)=0$. We see that $xf(x)$ is 0 at 0 and 1 and is positive in between so it must have a maxima. Is it possible that $xf(x)$ has more than one maxima on $[0,1]$?
This question came from trying to think of simple examples to test some ideas I'm working on and a few minutes of mucking around with polynomial interpolation etc showed I'm not very good at coming up with counterexamples. So if you have any comments on that let me know.
Yes
There are probably simpler examples, but $$f(x)= \frac{\sin (6 \pi x) }{24} + 1 - x$$ seems to be strictly decreasing on $[0,1]$ but $xf(x)$ appears to have two maxima in $[0,1]$.