Let $f\in C^1[a,b]$, $f(a)=0$ and suppose that $\lambda\in\mathbb R$, $\lambda >0$, is such that $$|f'(x)|\leq \lambda |f(x)|$$for all $x\in [a,b]$. Is it true that $f(x)=0$ for all $x\in [a,b]$?
Solution:
Say there exits an interval $[c,d]$ where funtion $f(x)$ is non zero so $|f(x)|$ is differentiable in $[c,d]$ with $f(c)=0$. Now consider the function $g(x)=e^{-\lambda x}|f(x)|$ and see that $g'(x)\leq 0$. So $g $ is decreasing. So, $x\geq a \implies g (x)\leq g(a)=0$. Hence, $|f (x)|\leq 0\implies f (x)=0$
Is this solution correct? basically i am asking is the differentiablity of $|f(x)|$ makes sense
