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

Answered by mindispower last updated on 15/Jul/21

$$\iff \frac{y(x+y)+z(x+z)}{\sqrt{(x+y)(x+z)}}⩾y+z....$$$$y(x+y)+z(x+z)=x(y+z)+y^2+z^2⩾x(y+z)+\frac{1}{2}{(y+z)}^2$$$$=(y+z)(x+\frac{1}{2}(y+z))=(y+z)\left(\frac{1}{2}((x+y)+(x+z))\right)$$$$⩾(y+z).\frac{1}{2}.2\sqrt{(x+y)(x+z)}=(y+z)\sqrt{(x+y)(x+z)}$$$$\Rightarrow \frac{y(x+y)+z(x+z)}{\sqrt{(x+y)(x+z)}}⩾\frac{(y+z)\sqrt{(x+y)(x+z)}}{\sqrt{(x+y)(x+z)}}=y+z$$$$$$$$$$$$$$

Commented by mathdanisur last updated on 16/Jul/21

$$thankyouSer$$

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