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Math Help - How can i show that ...

  1. #1
    Member
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    Question How can i show that ...

    If a + b + c + d = 1 \Rightarrow

    Show that \sqrt{(4a + 1)} + \sqrt{(4b + 1)} + \sqrt{(4c + 1)} + \sqrt{(4d + 1)} \le 6
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  2. #2
    Junior Member Infophile's Avatar
    Joined
    May 2009
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    Hello,

    Maybe I have an idea :

    a+b+c+d=1\Leftrightarrow (4a+1)+(4b+1)+(4c+1)+(4d+1)=8

    And if we write \left\{\begin{array}{l}x=4a+1\\y=4b+1\\z=4c+1\\t=4  d+1\end{array}\right. we have to prove that :

    x+y+z+t=8\Longrightarrow \sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{t}\le 6

    The function x\mapsto \sqrt{x} is concave, so Jensen's inequality gives :

    \frac{\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{t}}{4}\le \sqrt{\frac{x+y+z+t}{4}}=\sqrt{\frac{8}{4}}=\sqrt{  2}

    Hence : \color{red}\boxed{\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt  {t}\le 4\sqrt{2}<6}

    Last edited by Infophile; May 8th 2009 at 04:05 PM.
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