Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .
Solution 1:
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.