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Inequalities
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June 15, 2021 11:30 am
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(@armor)
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a,b,c,d are positive numbers, show  \sqrt{(a+c)(b+d)}\geq \sqrt{ab}+\sqrt{cd}.

 

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June 15, 2021 11:30 am
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

\small ad+bc\geq 2\sqrt{abcd}\Leftrightarrow ad+bc+ab+cd\geq ab+2\sqrt{abcd}+cd\Leftrightarrow (a+c)(b+d)\geq (\sqrt{ab}+\sqrt{cd})^2\Leftrightarrow \sqrt{(a+c)(b+d)}\geq \sqrt{ab}+\sqrt{cd}

since a,b,c,d > 0.

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