Let $f$ be continuous on $[a,b]$, with continuous first and second derivatives on $[a,b]$. Suppose $f$ has at least 3 distinct zeroes in $[a,b]$. Show that $f''(x)+g(x)f'(x)-f(x)=0$ has at least one root in $[a,b]$, where $g$ is any continuous function over $[a,b]$.
Say $f(\alpha)=f(\beta)=f(\gamma)=0;\;\alpha,\beta,\gamma\in(a,b)$ and $\alpha<\beta<\gamma$ since they are distinct.
Hence, by Mean Value Theorem, I know $f'(\zeta_1)=0$ where $\zeta_1\in(\alpha,\beta)$; $f'(\zeta_2)=0$ where $\zeta_2\in(\beta,\gamma)$. And also for $f''(\zeta_3)=0$; where $\zeta_3\in(\zeta_1,\zeta_2)$.
I know I have to use intermediate value theorem, but I have no idea on how to use my result to do that. Thank you.