1 reply to this topic

[hoahoanaa]

cho a,b,c là các số thực dương chung minh rang

$\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}}\geq 2(\sqrt{\frac{c}{a+b}}+\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{a+c}})$

[iloveyouproht]

cho a,b,c là các số thực dương chung minh rang

$\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}}\geq 2(\sqrt{\frac{c}{a+b}}+\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{a+c}})$

Ta có :

VT=$\sum \sqrt{\frac{a+b}{c}}\geq \sum \sqrt{\frac{(\sqrt{a}+\sqrt{b})^{2}}{2c}} \geq \sum \frac{\sqrt{a}+\sqrt{b}}{\sqrt{2c}}=\sum \sqrt{a}(\frac{1}{\sqrt{2b}}+\frac{1}{\sqrt{2c}})\geq \sum \sqrt{a}\frac{4}{\sqrt{2}(\sqrt{c}+\sqrt{b})}\geq \sum \sqrt{a}\frac{4}{2\sqrt{b+c}}=2(\sum \sqrt{\frac{a}{b+c}})$=VP=> ĐPCM

Edited by iloveyouproht, 11-01-2017 - 10:31.