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x,y,z为正实数 x/(2x+y+z)+y/(x+2y+z）+z/(x+y+2z)

分类: 作业答案 • 2021-12-30 13:14:48
问题描述：

x,y,z为正实数 x/(2x+y+z)+y/(x+2y+z）+z/(x+y+2z)

答

x/(2x+y+z)=[3*(2x+y+z)-(x+2y+z）-(x+y+2z)-]/(4(2x+y+z))
y/(x+2y+z)=[3*(x+y+2z)-(2x+y+z)-(x+y+2z)]/(4(x+2y+z）)
z/(x+y+2z)=[3*(x+y+2z)-(2x+y+z)-(x+2y+z)]/(4(x+2y+z）)
所以
左边=（3/4）*3 -（1/4）（(2x+y+z)/(x+2y+z）+(x+2y+z）/(2x+y+z)+(x+2y+z）/(x+y+2z)+(x+y+2z)/(x+2y+z）+(x+y+2z)/(2x+y+z)+(2x+y+z)(x+y+2z)）
又因为（(2x+y+z)/(x+2y+z）+(x+2y+z）/(2x+y+z)+(x+2y+z）/(x+y+2z)+(x+y+2z)/(x+2y+z）+(x+y+2z)/(2x+y+z)+(2x+y+z)(x+y+2z)）>=6
所以左边

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