TITLE: Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .
QUESTION [5 upvotes]: Prove that if $f$ is differentiable on $ [a,b]$ and if $ f(a)=f(b)=0$ then for any real $k$ there is an $ x \in (a,b) $such that $$kf(x)+f'(x)=0 $$ 
As all the conditions of Rolle's theorem are satisfied one can say that there is at least one  $c \in (a,b) $ such that $f'(c) =0$
How should I proceed furthur ? How can I use this to get to the required equation ?

REPLY [5 votes]: Consider the function $h(x)=e^{kx}f(x)$.  Apply Rolle's Theorem on $h(x)$ for the interval $[a,b]$
EDIT:
When I have to show something involving derivative of a function when I have some information about the function, I usually try to integrate the given expression. Doing that usually gives me a hint about what I have to do or suppose or assume. Like in this case if you divide by $f(x)$ and integrate you will be able to figure out the supposition.