1. ## A proof about differentiable functions

Hello

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...

2. Originally Posted by moses
Hello

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...
I think that you have made a good start there. If there are two vanishing points of f without a vanishing point of g between them, then (g(x))^2 will be strictly positive throughout that interval. Also, $\displaystyle 0\ne \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} = \frac d{dx}\Bigl(\frac{f(x)}{g(x)}\Bigr).$ Now show that that contradicts Rolle's theorem for the function f/g.

3. Oh. It seems so obvious now. Thanks.