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Math Help - Proving an inequality using the Mean Value Theorem

  1. #1
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    Proving an inequality using the Mean Value Theorem

    I am trying to prove the following inequality using the Mean Value Theorem:

    \ln (3x) \leq 3x - 1

    I can do it other ways but the question says to use the Mean Value Theorem.

    I have tried applying the Mean Value Theorem to the function f(x) = \ln (3x) on the interval \left[\frac{1}{3}, \, x\right] but cannot get the required inequality.

    If anyone can help I would be grateful.
    Last edited by mr fantastic; August 22nd 2010 at 02:47 AM. Reason: Fixed an obvious typo.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Apply the Mean Value Theorem to the function f(x)=\ln(3x)-3x+1, \ x>0 on the intervals \left[\displaystyle\frac{1}{3},x\right] and \left[ x,\displaystyle\frac{1}{3}\right].
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  3. #3
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    Quote Originally Posted by red_dog View Post
    Apply the Mean Value Theorem to the function f(x)=\ln(3x)-3x+1, \ x>0 on the intervals \left[\displaystyle\frac{1}{3},x\right] and \left[ x,\displaystyle\frac{1}{3}\right].
    Thankyou, very helpful.
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