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x2y2+z2y2+y2x2大于等于xyz(x+y+z)x+y+z=2求证x2y2+z2y2+y2x2大于等于xyz(x+y+z)
题目详情
x2y2+z2y2+y2x2大于等于xyz(x+y+z)
x+y+z=2 求证
x2y2+z2y2+y2x2大于等于xyz(x+y+z)
x+y+z=2 求证
x2y2+z2y2+y2x2大于等于xyz(x+y+z)
▼优质解答
答案和解析
∵x²y²+y²z²≥2y²xz
z²y²+x²z²≥2z²xy
x²z²+x²y²≥2x²zy
以上3式左边相加与右边相加
∴2(x²y²+y²z²+z²x²)≥2y²xz+2z²xy+2x²zy=2xyz(x+y+z)
∴(x²y²+y²z²+z²x²)≥xyz(x+y+z)
z²y²+x²z²≥2z²xy
x²z²+x²y²≥2x²zy
以上3式左边相加与右边相加
∴2(x²y²+y²z²+z²x²)≥2y²xz+2z²xy+2x²zy=2xyz(x+y+z)
∴(x²y²+y²z²+z²x²)≥xyz(x+y+z)
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