作业帮已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]
来源：学生作业帮助网 编辑：作业帮 时间：2024/10/03 17:45:36
![已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]](/uploads/image/z/1773004-4-4.jpg?t=%E5%B7%B2%E7%9F%A5a%2Cb%2Cc%E5%B1%9E%E4%BA%8ER%2B%2C%E6%B1%82%E8%AF%81%EF%BC%9A3%5B%28a%2Bb%2Bc%29%2F3-%E4%B8%89%E6%AC%A1%E6%A0%B9%E5%8F%B7%E4%B8%8B%EF%BC%88abc%29%5D%E2%89%A52%5B%28a%2Bb%29%2F2-%E6%A0%B9%E5%8F%B7%E4%B8%8B%28ab%29%5D%E5%B7%B2%E7%9F%A5a%2Cb%2Cc%E5%B1%9E%E4%BA%8ER%2B%2C%E6%B1%82%E8%AF%81%EF%BC%9A3%5B%28a%2Bb%2Bc%29%2F3-%E4%B8%89%E6%AC%A1%E6%A0%B9%E5%8F%B7%E4%B8%8B%EF%BC%88abc%29%5D%E2%89%A52%5B%28a%2Bb%29%2F2-%E6%A0%B9%E5%8F%B7%E4%B8%8B%28ab%29%5D)
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已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]
已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]
已知a,b,c属于R+,求证：
3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]
已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]已知a,b,c属于R+,求证：3[(a+b+c)/3-三次根号下（abc)]≥2[(a+b)/2-根号下(ab)]
原不等式整理后即证
c+2(ab)^(1/2)≥3(abc)^(1/3)
又由均值不等式知：
左边=c+(ab)^(1/2)+(ab)^(1/2)
≥3[c*(ab)^(1/2)*(ab)^(1/2)]
=3(abc)^(1/3)
=右边
得证
c+2(ab)^(1/2)≥3(abc)^(1/3)
又由均值不等式知：
左边=c+(ab)^(1/2)+(ab)^(1/2)
≥3[c*(ab)^(1/2)*(ab)^(1/2)]
=3(abc)^(1/3)
=右边