A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?
Solution 1:
Let $g(x) = e^{-cx}f(x)$ then $g(x) \geq 0$ for all $x \in [0, 1]$. Now $g'(x) = e^{-cx}\{f'(x) - cf(x)\} \leq 0$. So $g(x)$ is decreasing and $g(0) = 0$ therefore $g(x) \leq 0$ for all $x \in [0, 1]$. It now follows that $g(x) = 0$ for all $x \in [0, 1]$ and hence $f(x) = e^{cx}g(x) = 0$ for all $x \in [0, 1]$.
Its an old popular question and has also been solved on this website earlier but I am not able to find the reference. I had posted a bit difficult solution on my blog post (see problem 2 there).