Show that if $f$ is Positive-real function then $f=0$

Question

Let $f:[0,1] \to \mathbb{C}$ differentiable with $f(0)=0$ and :

$$\forall x \in [0,1]: \quad |f'(x)| \leq M|f(x)|\quad ( M\geq 0) $$

• Show that if $f$ is Positive-real function then $f=0$

I think that Lipschitz continuity and Rolle's theorem can be useful here but i don't know how to use them here if $f$ is Positive-real function then $$\forall x \in [0,1]\quad -Mf(x)\leq f'(x)\leq Mf(x) $$ which let us to write them in the following expression:

\begin{cases} M\exp(Mx)f(x)+\exp(Mx)f'(x)\geq 0 \\ -M\exp(-Mx)f(x)+\exp(-Mx)f'(x)\leq 0\end{cases}