I did not know what to search to see if this existed elsewhere. But, I could not find it. Here's the question:
Does there exist a continuously differentiable function, $f: [0,1] \rightarrow [0, \infty)$, such that the following hold?
(i) $f(0) = 0$
(ii) $f'(x) \le c f(x)$ for all $x$ ($c$ is a fixed constant)
(iii) $f \not\equiv 0$
This was a fun problem someone asked me a long time ago. I have always been convinced there is no such function, but I cannot prove it. I have mainly tried using the mean value theorem (in multiple forms), and re-wording the problem in terms of the integral of a continuous function to no avail. I spent some time trying to use (ii), the definition of derivative, and squeeze theorem, but no luck. After much time trying to prove non-existence, I spent a little time trying to find such a function... also with no luck. This problems ability to avoid being solved has since taken away the original fun and replaced it with a twinge of annoyance.