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Math Help - Schwarz's inequality

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    Schwarz's inequality

    A) Given f(x) = ax^2 + 2bx + c with a>0. By considering the minimum, prove that f(x)>=0 for all real x if and only if b^2-ac<=0.

    B) From A, let f(x)= (a1x+b1)^2 + (a2x+b2)^2 + ...... +(anx + bn)^2 and deduce Schwarz's inequality: (a1b1 + a2b2 + .... +anbn)^2 <= (a1^2 + a2^2 + ..... + an^2)(b1^2 + b2^2 + ..... + bn^2)

    C) Show that equality holds in Schwarz's inequality only if there exists a real number x that makes aix equal to -bi for every value of i from 1 to n. (the i's are subs)
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  2. #2
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    Quote Originally Posted by mistykz View Post
    A) Given f(x) = ax^2 + 2bx + c with a>0. By considering the minimum, prove that f(x)>=0 for all real x if and only if b^2-ac<=0.

    B) From A, let f(x)= (a1x+b1)^2 + (a2x+b2)^2 + ...... +(anx + bn)^2 and deduce Schwarz's inequality: (a1b1 + a2b2 + .... +anbn)^2 <= (a1^2 + a2^2 + ..... + an^2)(b1^2 + b2^2 + ..... + bn^2)
    Part A is easy, you should know this fact.

    Notice that f(x)\geq 0 for any x\in \mathbb{R} because (a_ix+b_i)^2 \geq 0 \mbox{ for }1\leq i\leq n.
    But, this is a quadradic function:
    f(x) = \left( \sum_{i=1}^n a_i^2 \right)x^2 + 2\left( \sum_{i=1}^n a_ib_i \right) x + \left( \sum_{i=1}^n b_i^2 \right) .
    But above if is necessary and sufficient for b^2 - 4ac\geq 0, thus:
    4\left( \sum_{i=1}^n a_ib_i \right)^2-4\left( \sum_{i=1}^n a_i^2 \right)\left( \sum_{i=1}^n b_i^2 \right)\geq 0.
    Q.E.D.
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