Should $f(x) \equiv 0$ if $0\le f'(x)\le f(x)$ and $f(0)=0$? Assume $f(x)$ is a real-function defined on $[0,+\infty)$ and satisfies the followings:


*

*$f'(x) \geq 0$

*$f(0)=0$

*$f'(x) \leq f(x)$
Should we always have $f(x) \equiv 0$ ? Thanks for any solution.

Let $x_0\in[0,x]$ be a point in which $f'$ assumes a maximum. Then 
$$f(x) = f(x) - f(0) = \int_0^xf'(t)dt \le xf'(x_0) \le xf(x_0) \le xf(x)$$
so $f$ must be $0$ on $[0,1)$ and by continuity $f(1) = 0$. Since $f(x - 1)$ satisfies properties 1), 2), 3), the result follows by induction.
