TITLE: How derivative relates to roots of original function
QUESTION [5 upvotes]: Assume $f$ is differentiable on $\mathbb{R}$. Show that for any $ k \in \mathbb{R}$, $f' + kf$ has a root between any two distinct roots of $f$. 
I am completely stumped on this. What are some good ways to show existence of zeros? Thanks.

REPLY [5 votes]: Here's a solution (I asked someone for help with it):
Let $\alpha$ and $\beta$ be two distinct roots of $f$. Consider $g = e^{kx}f$. Then note that $g' = ke^{kx}f + e^{kx} f'$. By Rolle's theorem, since $g(\alpha)=g(\beta)=0$, there exists a $c \in (\alpha, \beta)$ such that $g'=0$ $\implies$ $f'(c)+kf(c)=0$.