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Prove an inequality

  1. Apr 18, 2015 #1
    Is it possible to prove this:
    x^x + y^y < (x+y)^(x+y) for every x,y >=1 ?
     
  2. jcsd
  3. Apr 18, 2015 #2

    RaulTheUCSCSlug

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    Gold Member

    Well, let me think, since I am not really sure if this would work for all parameters of x and y.... never mind, you have x,y >1! What you could do is take the derivative of both equations to measure it's change in slope, and if (x+y)^(x+y)is greater, then it will have a change in slope that is greater then the other equation. But I am not sure if that is what you want.
     
  4. Apr 18, 2015 #3
    Iam just proving it,just expand the Right side (x+y)^(x+y), u get x multiplied by x+y times which is obviously greater than x^x and same in case of y and remaining terms of expansion are positive as x,y>1 and no negative terms in expansion. hope it helps.
     
  5. Apr 18, 2015 #4
    Thanks for the answers, but I would prefer an algebraic solution.
    I did what Raul said with the graph but I would like an algebraic sol.
    I believe that it is solved algebraically and it is not a transcendental equation.
     
    Last edited: Apr 18, 2015
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