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Math Help - Proof

  1. #1
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    Proof

    I need to prove:

    for a,b,c > 0

    prove a/(b+c) + b/(a+c) + c/(b+a) is less than or equal to 3/2
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by Nichelle14
    I need to prove:

    for a,b,c > 0

    prove a/(b+c) + b/(a+c) + c/(b+a) is less than or equal to 3/2
    it can be proved by using marvelous AM-GM inequality of positive real numbers.

    let x=a/(b+c) + b/(a+c) + c/(b+a)
    now, b+c>=2(bc)^(1/2)
    hence,1/(b+c)<=1/2(bc)^(1/2)
    hence,2x<=a/(bc)^(1/2) + b/(ac)^(1/2) + c/(ba)^(1/2)
    again applying AM-GM inequality on the right hand side of the equation
    2x<=3
    x<=3/2
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  3. #3
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    AM/GM is your friend. Here's another way, a bit longer but with a couple of useful tricks.

    Firstly observe that \frac1X + \frac1Y + \frac1Z \ge 3\sqrt[3]{\frac1{XYZ}} = 3 \big/ \sqrt[3]{XYZ} \ge \frac{9}{X+Y+Z}, using AM/GM twice.

    In your case this says that \frac a{b+c} + \frac b{c+a} + \frac c{a+b} \ge 9\big/\big(\frac{b+c}a + \frac{c+a}b+\frac{a+b}c\big). I claim that this last sum is \ge 6. We can divide through by c and then we need to prove that \frac{b+1}a + \frac{1+a}b+a+b \ge 6. Fix b and treat as a function of a. It is minimised when the derivative = 0 and this is at b^2 = a and the value is a^2 + 1/a^2 + 2(a+1/a). A further application of AM/GM shows that F+1/F \ge 2 whatever F is, so the minimum is always \ge 6, as claimed.
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  4. #4
    Super Member malaygoel's Avatar
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    Quote Originally Posted by Nichelle14
    I need to prove:

    for a,b,c > 0

    prove a/(b+c) + b/(a+c) + c/(b+a) is less than or equal to 3/2
    I think that you have posted a wrong question and I have proved a wrong inequality.
    Let a=1, b=2 ,c=3
    then
    \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{b+a}=\frac{1}{5} + \frac{2}{4} + \frac{3}{3} >3/2
    I have started a new thread 'Proof on inequality'in "pre algebra and algebra" asking for proof of the inequality \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{b+a}\geq \frac{3}{2}.
    Could someone please help how I proved an inequality which does not exist?
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