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Math Help - Abstract Inequalities

  1. #1
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    Abstract Inequalities

    Q. Having some trouble with this one. Can anyone help me out?


    Many thanks.


    Q. Use the result, a^2+b^2\geq2ab, to prove that a^2+b^2+c^2\geq ab+bc+ca for all real values of a, b & c.


    Attempt: \frac{a^2+b^2}{2}\geq ab
    if \frac{a^2+b^2}{2}-ab\geq0
    if \frac{(a+b)(a-b)}{2}-ab\geq0
    if \frac{a^2-ab+ab+b^2}{2}-ab\geq0
    if \frac{a^2-ab+ab+b^2-2ab}{2}\geq0
    if \frac{a^2-2ab+b^2}{2}
    if \frac{(a-b)^2}{2}\geq0


    Thus: a^2+b^2+c^2\geq ab+bc+ca
    if \frac{2a^2+2b^2+2c^2}{2}-ab-bc-ca\geq0
    if \frac{2a^2+2b^2+2c^2-2ab-2bc-2ca}{2}\geq0
    if a^2-b^2-c^2-ab-bc-ca\geq0
    if (a+b+c)^2\geq0
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  2. #2
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    Re: Abstract Inequalities

    Quote Originally Posted by GrigOrig99 View Post
    Q.
    Q. Use the result, a^2+b^2\geq2ab, to prove that a^2+b^2+c^2\geq ab+bc+ca for all real values of a, b & c.
    \begin{align*} a^2+b^2 &\ge 2ab \\ a^2+c^2 &\ge 2ac \\ b^2+c^2&\ge 2bc \\ \text{Add all three  }2a^2+2b^2+2c^2 &\ge 2ab+2ac+2bc \end{align*}
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  3. #3
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    Re: Abstract Inequalities

    Great, thank you very much.
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