I've been stuck on the following math question:
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ have two derivatives with $f(0) = 0$ and $f'(x) \leq f(x)$ for all $x$. Is $f(x) = 0$ for all $x$?
I've tested several functions, and I believe the answer is true. I have no clue about how to prove this statement though.
For example, if we have a constant function $f(x) = c$, then we must have $c = 0$ due to the $f(0) = 0$ condition. I've also tried polynomial and trignometric functions, and I cannot find a counterexample.
I thought about somehow using the Mean Value Theorem or Rolle's Theorem, but I didn't get anywhere with either of those.