This is a question from a sample paper of a entrance exam. I did this type of problems using MVT but this time i dont have any hint how to solve this The question is "let $ f :[0,1]- R $ is differentiable and $ f(0)=0. $ Given that there exist a $c\in (0,1)$ such that $ |f'(x)| \leq c|f(x)|$ for all $ x \in [0,1]. $ Then prove that $ f(x) =0$ for all $x\in [0,1].$
I was thinking if i can prove this by contradiction. That is if I assume $ f(x)>0 $ or $ f(x)<0 $ then by MVT i can arrive at a contradiction. But that's where i am stuck now since $c\in (0,1) $ then $c<1$ then $|f'(x)|<|f(x)|$ but i dont know what to do after that. Next i have some idea if i can show that $f'(x) = 0$ for all $x $ then $ f $ is a constant function and since $f(0)=0 $ then the whole function is zero . That's why i need some hints to slove this problem.