Let $f(x),g(x)$ be differential functions, and $f'(x)g(x)\neq f(x)g'(x)$ for all $x\in\mathbb R$. Prove that between two roots of $f(x)$ there is a root of $g(x)$.
I guess this has to do with Rolle's theorem. I saw that when $f'(x)=0$, $g(x)\neq0$ and when $f(x)=0$, $g'(x)\neq0$, but I didn't manage to prove the conjecture. Thanks for any help!
Best Answer
Suppose that $a,b$ are roots of $f$ with $a<b$ and $g(x)\neq0$ for $x\in (a,b)$.
Consider the function $h(x)=\dfrac{f(x)}{g(x)}$ in $[a,b]$ to derive a contradiction
(is well defined, differentiable on [a,b] with $h(a)=h(b)$...).