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Experience with proofs

  1. Jul 24, 2008 #1
    1. The problem statement, all variables and given/known data

    For U(w)=sqrt(w), prove that U(pie(x)+(1-pie)y) > pie*U(x)+(1-pie)U(y)



    2. Relevant equations

    sqrt(x)=x^(1/2)

    3. The attempt at a solution

    I have:

    sqrt(pie(x)+(1-pie)y) > pie*sqrt(x)+(1-pie)sqrt(y) so...

    (pie(x)+(1-pie)y)^(1/2) > pie*(x)^1/2+(1-pie)(y)^1/2

    From here I don't know where to go. I don't have much experience with proofs so can anyone give some guidance? Thanks for your help!
     
    Last edited by a moderator: Jan 7, 2014
  2. jcsd
  3. Jul 27, 2008 #2
    Re: proof

    Try squaring both sides.
     
  4. Jul 27, 2008 #3
    Re: proof

    alright, then (pi(x)+(1-pi)y) > pi^2*(x)+(1-pi)^2*(y). Therefore pi+(1-pi)>(pi^2)+((1-pi)^2)
    =1 > (pi^2)+((1-pi)^2)
    =1-(pi^2)>(1-pi)^2

    Is this sufficient? What I don't quite get is that if you then square both sides, don't you get sqrt((1-(pi^2))) > sqrt((1-pi)^2)

    =sqrt(1)-pi > (1-pi)
    =1-pi > 1-pi ?? This obviously can't be right, so where did I go wrong? Thanks again.
     
  5. Jul 28, 2008 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: proof

    Here "pi" is NOT the usual "pi", it is a number between 0 and 1, right? This is essentially a proof that the square root function is concave.
     
  6. Jul 28, 2008 #5
    Re: proof

    yes exactly, sorry for not making that clear before.
     
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