5 (10) Suppose that \( f:[0,1] \to \mathbb{R} \) is continuous, \( f \) is differentiable on \( (0,1) \), and \( f(0) = 0 \). Assume that \( |f'(x)| \leq |f(x)| \), \( 0 < x < 1 \). Prove that \( f(x) = 0 \) for all \( x \in [0,1] \).