I'm trying to prove that if two functions f and g are differentiable on R
and they satisfy for all x
f'(x) * g(x) != f(x) * g'(x)
then between every two vanishing points of f there is a vanishing point of g.
I've tried to assume contrarily that there are two vanishing points of f without a vanishing point of g between them or on them (call them a and b), define a function f/g, and use Rolle's theorem to show that there exists a c in [a,b] such that f'(c)/g'(c) = 0. But that hasn't really helped...