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Thread: If a, b and c are positive real numbers, prove that 4(a + b) ≥ (a + b)

  1. #1
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    If a, b and c are positive real numbers, prove that 4(a + b) ≥ (a + b)

    If a, b and c are positive real numbers, prove that 4(a + b) ≥ (a + b) & that 9(a + b + c) ≥ (a + b + c)
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  2. #2
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    Quote Originally Posted by Mozart View Post
    If a, b and c are positive real numbers, prove that 4(a + b) ≥ (a + b) & that 9(a + b + c) ≥ (a + b + c)
    for the first one;
    $\displaystyle 4a^3+4b^3\geq a^3+3a^2b+3ab^2+b^3$
    $\displaystyle \Longleftrightarrow 3a^3+3b^3\geq 3a^2b+3ab^2$
    $\displaystyle \Longleftrightarrow a^3+b^3\geq a^2b+ab^2$
    $\displaystyle \Longleftrightarrow (a+b)(a^2-ab+b^2)\geq (a+b)(ab)$
    $\displaystyle \Longleftrightarrow a^2-ab+b^2\geq ab $
    $\displaystyle \Longleftrightarrow a^2-2ab+b^2\geq 0$
    $\displaystyle \Longleftrightarrow (a-b)^2\geq 0$
    Last edited by bram kierkels; Oct 2nd 2009 at 02:46 AM.
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  3. #3
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    that's wrong proof.

    to fix it, in each step you need to add an $\displaystyle \iff.$
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