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Math Help - A proof about differentiable functions

  1. #1
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    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...
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  2. #2
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    Quote Originally Posted by moses View Post
    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, 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.
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  3. #3
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    Oh. It seems so obvious now. Thanks.
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