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Math Help - Proof using Mean Value Theorem

  1. #1
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    Proof using Mean Value Theorem

    Hi guys,

    I'm a little bit confused over how to apply the Mean Value Theorem to this particular problem.

    Suppose f is a function such that f'(x)=1/x for all x>0. Prove that if f(1)=0, then f(xy)=f(x)+f(y) for all x,y >0.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Hweengee View Post
    Suppose f is a function such that f'(x)=1/x for all x>0. Prove that if f(1)=0, then f(xy)=f(x)+f(y) for all x,y >0.
    For y>0 define F_y(x) = f(xy)-f(x)-f(y) for x>0.
    Now F_y is differenciable on this integral and F'_y(x) = \frac{y}{yx} - \frac{1}{x} = 0.
    Thus, F'_y is konstant on (0,\infty) so F_y(x) = k for some k\in \mathbb{R}.
    But F_y(1) = 0 and so F_y(x) = 0.
    Thus, f(xy) = f(x)+f(y).
    Last edited by ThePerfectHacker; September 18th 2008 at 10:10 AM.
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  3. #3
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    " is konstant on so for some . "

    Do you mean F'_y(x) = 0 --> F_y(x) = k for some k in R? Sorry but I'm still a little confused.
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  4. #4
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    Quote Originally Posted by Hweengee View Post
    Do you mean F'_y(x) = 0 --> F_y(x) = k for some k in R? Sorry but I'm still a little confused.
    Sorry. I fixed it now. Does it make better sense?
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  5. #5
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    yes i get it now. thank you very much.
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