Proving $f(x)=0$

Question

Let $f$ be a function defined on $[a,b]$ so that $f$ is continuous on $[a,b]$, differential on $(a,b)$, $f(a)=0$ and there exist a real number $A$ so that $$|f'(x)| \leq A |f(x)|$$ for all $x \in [a,b]$

Prove that $f(x)=0$ for all $x \in [a,b]$

Answer 1

Assuming $f(x),f'(x)>0$, we can write

$$\frac{f'(x)}{f(x)}-A\le0$$ and by integration

$$\log f(x)-Ax-B=\log\frac{f(x)}{e^{AX+B}}$$ must be decreasing.

Many functions fulfill this condition.