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How to prove xy <= exp(x-1) +ylny using differentiation

  1. Sep 29, 2006 #1
    I am teaching a student in a course without partial differentiation so
    I can only think of the following method
    let b be a constant and set y=b so
    xb= exp(x-1) +blnb
    means b= exp(x-1) which means x = lnb+1
    now d(xb)/dx= b
    and d(exp(x-1)+blnb)/dx = exp(x-1)
    so if x > lnb+1
    xb> exp(x-1) + blnb
    since the Right hand side has a higher derivative

    so x > lnb+1 Implies xb> exp(x-1) + blnb

    so
    x > lny +1 implies xy< exp(x-1) + ylny

    Now replace x with b
    by = exp(b-1) + ylny
    means
    y=exp(b-1)
    now d(by)/dx= b
    and d( exp(b-1) + ylny)/dx = 1+ lny
    so if lny+ 1 > b
    by < exp(b-1) + ylny
    since the derivative of the right hand is larger

    so lny+1 >x implies xy < exp(x-1)+ylny
     
  2. jcsd
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