Show that $\exists c\in (a,b)$ such that $2f(c)+f'(c)=0$

Question

Let $f$ be a function continuous on $[a,b]$ and differentiable on $(a,b)$, $f(a) = f(b) = 0$

Show that $\exists c\in(a,b)$ such that $2f(c)+f'(c) = 0$

This is a problem on continuity and differentiability, and I have tried to use the Intermediate value theorem and Rolle's theorem with no success.

Any suggestions on how to solve it using the theorems above?

Answer 1

Hint: Consider $g(x)=f(x)e^{2x}$.