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Math Help - Problem concerning differentiation

  1. #1
    Senior Member Shanks's Avatar
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    Problem concerning differentiation

    f(x) is continous in [0,1], differentiable in (0,1), and f(0)=0,f(1)=1.
    prove that for any positive numbers a and b, there exist x_1 < x_2 in (0,1) such that
    \frac{a}{f'(x_1)}+\frac{b}{f'(x_2)}=a+b.
    how to prove this kind of problem about " there exist ..."?
    I know there is three theorems in the textbook may be helpful.
    But here there are two parameters.
    Any suggestion, hint or solution is greatly appreciated!
    Last edited by Shanks; December 13th 2009 at 07:15 PM.
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  2. #2
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    We can try the Mean Value Theorem:

    \mbox{There exists a }c, 0< c<1,\mbox{ such that }\frac{f(1)-f(0)}{1-0}=f'(c).
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    Senior Member Shanks's Avatar
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    I know there exist a number c in (0,1) such that f'(c)=1.
    that is obviously true by the theorem. And then.....
    we still didn't get closer to the result.
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  4. #4
    Senior Member Shanks's Avatar
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    Any body help me, PLZ!
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  5. #5
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    Quote Originally Posted by Shanks View Post
    I know there exist a number c in (0,1) such that f'(c)=1.
    that is obviously true by the theorem. And then.....
    we still didn't get closer to the result.
    This is the result: x_1=x_2=c.
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  6. #6
    Senior Member Shanks's Avatar
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    Sorry, I forgot the condition that x_1<x_2.
    here I made a complement.
    Now how to prove it? help!!!
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    Edit: Reworking
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