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Real Analysis proof continuity

  1. Nov 10, 2007 #1
    1. The problem statement, all variables and given/known data
    Suppose that the function f is continuous on [a,b] and X1 and X2 are in [a,b]. Let K1 and K2 be positive real numbers. Prove that there exist c between X1 and X2 for which

    f(c) = (K1f(X1) + K2f(X2))/(K1+k2)

    2. Relevant equations

    3. The attempt at a solution I know I am supposed to have attempted this before asking for help, but honestly I have looked at this problem for over an hour and cannot figure out what to use. I have tried looking at it in terms of Rolles Thrm, Intermediate Value Thrm, and Mean Value Thrm, but nothing is clicking. I also tried manipulating it just by moving things around. I don't seem to be getting anywhere and I'm just hoping someone can point me in the right direction. Then at least I can come up with a decent attempt.
  2. jcsd
  3. Nov 10, 2007 #2


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    Have you noticed that according the the intermediate value thm, all you need to do is to prove that

    f(X1) [itex]\leq[/itex] (K1f(X1) + K2f(X2))/(K1+K2) [itex]\leq[/itex] f(X2) ?

    Supposing f(X1) [itex]\leq[/itex] f(X2).
  4. Nov 11, 2007 #3
    oh god. I feel dumb now. I'm not even going to post back. I got it.
    Thank you.
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