Derivate problem

Let $f$ be continuous on $[a, b]$ and differentiable on $(a, b)$ with $f(c) = 0$ for some $c \in [a, b]$. If there exists $M \in \mathbb{R}$ such that $\vert f'(x) \vert \leq M \vert f(x) \vert$, $\forall x \in [a, b]$, then $f(x) = 0$, $\forall x \in [a, b]$.

The case $c = a$ is often studied in this context, For example[1][2][3][4]I am wondering if the result remains true if $c \neq a$.