Let $f: \mathbb R \to \mathbb R$ be function of class $C^{\infty}$ such, that $\forall x \le 0$ $f(x)=0$ and $\forall x > 0$ $f(x)>0$. Prove that $\forall B>0$ $\forall \varepsilon >0 $ $\exists n \in \mathbb N$ $\exists x \in ]0, \varepsilon[$ : $|f^{(n)} (x)| >B^n$.
I have no idea how to start. I'm expecting it will be necessary to use Taylor theorem in certain part of the proof but I don't now how to do it. I'd appreciate any advices.
