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If a, b, c, x, y, z are all positive, then prove that (ab + xy) . (ax + by) ≥ 4abxy.

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Applying AM - GM inequality, we have

\(\frac{ab+xy}{2}≥\sqrt{ab.xy}\)    ⇒   (ab + xy) ≥ 2\(\sqrt{ab.xy}\)                ......(i)

and \(\frac{(ab+xy)}{2}≥\sqrt{ab.xy}\)   ⇒   (ab + xy) ≥ 2\(\sqrt{ab.xy}\)         ......(ii)

Multiplying (i) and (ii), we get

(ab + xy) (ax + by) > 4\(\sqrt{abxy}\sqrt{abxy}\)   ⇒  (ab + xy) (ax + by) > 4 abxy.

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