Chủ đề này có 1 trả lời

[Gammaths11]

CMR: A=$\sqrt{(x+y)(y+z)(z+x)}.\left ( \frac{\sqrt{y+z}}{x}+\frac{\sqrt{z+x}}{y}+\frac{\sqrt{x+y}}{z} \right )\geq 4\sqrt{2}$

[Gammaths11]

A=$\sum \left ( y+z \right )\sqrt{\frac{(x+y)(z+x)}{x^{2}}}=\sum (y+z)\sqrt{\frac{x^{2}+x(y+z)+yz}{x^{2}}}\geq \sum (y+z)\sqrt{\frac{x^{2}+2\sqrt{yz}.x+yz}{x^{2}}}=\sum (y+z)(1+\frac{\sqrt{yz}}{x})=\sum (y+z)+\frac{(y+z)\sqrt{yz}}{x}\geq \sum y+z+\frac{2yz}{x}=2(x+y+z)+(\frac{xy}{z}+\frac{yz}{x})+(\frac{yz}{x}+\frac{xz}{y})+(\frac{xz}{y}+\frac{xy}{z})\geq 2\sqrt{2}+2(x+y+z)=4\sqrt{2}$