Question-146817

Question Number 146817 by mathdanisur last updated on 15/Jul/21

Answered by mindispower last updated on 15/Jul/21

\Leftrightarrow \frac{y (x + y) + z (x + z)}{\sqrt{(x + y) (x + z)}} ⩾ y + z \dots .

y (x + y) + z (x + z) = x (y + z) + y^{2} + z^{2} ⩾ x (y + z) + \frac{1}{2} {(y + z)}^{2}

= (y + z) (x + \frac{1}{2} (y + z)) = (y + z) (\frac{1}{2} ((x + y) + (x + z)))

⩾ (y + z) . \frac{1}{2} . 2 \sqrt{(x + y) (x + z)} = (y + z) \sqrt{(x + y) (x + z)}

\Rightarrow \frac{y (x + y) + z (x + z)}{\sqrt{(x + y) (x + z)}} ⩾ \frac{(y + z) \sqrt{(x + y) (x + z)}}{\sqrt{(x + y) (x + z)}} = y + z

Commented by mathdanisur last updated on 16/Jul/21

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