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Math Help - Prove that this function is increasing

  1. #1
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    Prove that this function is increasing

    We are given, for a,b>0

    f(a,b)=\frac{a-b}{ln a-ln b}  \ \ \ if     a \neq b
    and
    f(a,b)=a if a=b

    prove that f(a,b) is strictly increasing in both a and b


    I take the derivative, but I can't prove that it is positive. Help is much appreciated
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by altave86 View Post
    We are given, for a,b>0

    f(a,b)=\frac{a-b}{ln a-ln b}  \ \ \ if     a \neq b
    and
    f(a,b)=a if a=b

    prove that f(a,b) is strictly increasing in both a and b


    I take the derivative, but I can't prove that it is positive. Help is much appreciated
    b<\frac{a-b}{\ln(a)-\ln(b)}, see this using the MVT on f(x)=\ln(x).
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  3. #3
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    Quote Originally Posted by altave86 View Post
    We are given, for a,b>0

    f(a,b)=\frac{a-b}{ln a-ln b} \ \ \ if a \neq b
    and
    f(a,b)=a if a=b

    prove that f(a,b) is strictly increasing in both a and b


    I take the derivative, but I can't prove that it is positive. Help is much appreciated

    \frac{df(a,b)}{da}=\frac{\ln a - \ln b -1 +\frac{b}{a}}{(\ln a-\ln b)^2}=\frac{\ln(a\slash b)+b\slash a -1}{(\ln a-\ln b)^2}

    Since \ln x+x^{-1}>=1\,\,\,\forall\,x>0 (check the minimal point of f(x):=\ln x+\frac{1}{x}-1) , we're done.
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