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Proof using cauchy schwerz inequality algorithm

If a,b,c are three positive numbers then using cauchy schwerz inequality prove that

(a+b+c)(1a+1b+1c)32

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Consider two vectors

$\begin{align} u &= ( \sqrt a , \sqrt b, \sqrt c )\\ v &= (\frac{1}{\sqrt a} , \frac{1}{\sqrt b}, \frac{1}{\sqrt c})\\ u.v &= ( \sqrt a , \sqrt b, \sqrt c ) (\frac{1}{\sqrt a} , \frac{1}{\sqrt b}, \frac{1}{\sqrt c})\\ u.v &= (\frac{ \sqrt a}{\sqrt a} +\frac{\sqrt b}{\sqrt …

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