Do derivative always coincide with the function at some point?

Question

Given f is differentiable on $(a,b)$, continuous on $[a,b]$ and f(a)=f(b)=0, then is this true that $f'(c)=f(c)$ for some $c\in(a,b)$?. Intuitively i feel this statement is true, so I tried to use mean value theorem for some combination of functions like adding and composing f with other functions etc, but i am unable to prove.

Answer 1

This follows from Rolle's theorem applied to the function $-e^{-x} f(x)$ which has derivative $$e^{-x} (f(x) - f'(x)).$$