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Question:

pavelpimen
pavelpimen
Student
High School - 10th Grade

f(x)=ln x - [2(x-1)/x+1]. Prove that if x>1, then

 ln x > [2(x-1)/x+1]

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Posted by pavelpimen on Monday April 6, 2009 at 7:06 AM and tagged with function, math.

Answers:

  1. giorgiana1976
    giorgiana1976 Teacher
    Doctorate

    eNotes Editor

    Best answer as selected by question asker.

    We've noticed that if we bring the paranthesis 2(x-1)/(x+1) on the left side of the inequality, which has to be demonstrated, is no one else but the function f(x) itself.

    So, we have to demonstrate that f(x)>0. For proofing this, we have to verify if the first derivative of the function is also positive.

    f'(x)=[lnx-2(x-1)/(x+1)]'=

    =(1/x)-{[2(x-1)'*(x+1)-2(x-1)*(x+1)']/(x+1)^2}=

    =(1/x)-(2x+2-2x+2)/(x+1)^2=(1/x)-(4)/(x+1)^2=

    =[(x+1)^2-4x]/x*(x+1)^2=(x^2+2x+1-4x)/x*(x+1)^2=

    =(x^2-2x+1)/x*(x+1)^2=(x-1)^2/x*(x+1)^2

    But for x>1, (x-1)^2>0 and x*(x+1)^2>0, so f(x)>0

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    Posted by giorgiana1976 on Monday June 1, 2009 at 11:07 AM