已知a,b,c∈正实数,a+b+c=1.求证:1/(根号a+根号b)+1/(根号b+根号c)+1/(根号c+根号a)≥(
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已知a,b,c∈正实数,a+b+c=1.求证:1/(根号a+根号b)+1/(根号b+根号c)+1/(根号c+根号a)≥(3根号3)/2
![已知a,b,c∈正实数,a+b+c=1.求证:1/(根号a+根号b)+1/(根号b+根号c)+1/(根号c+根号a)≥(](/uploads/image/z/5291151-15-1.jpg?t=%E5%B7%B2%E7%9F%A5a%2Cb%2Cc%E2%88%88%E6%AD%A3%E5%AE%9E%E6%95%B0%2Ca%2Bb%2Bc%3D1.%E6%B1%82%E8%AF%81%EF%BC%9A1%2F%28%E6%A0%B9%E5%8F%B7a%2B%E6%A0%B9%E5%8F%B7b%29%2B1%2F%28%E6%A0%B9%E5%8F%B7b%2B%E6%A0%B9%E5%8F%B7c%29%2B1%2F%28%E6%A0%B9%E5%8F%B7c%2B%E6%A0%B9%E5%8F%B7a%29%E2%89%A5%28)
首先,由Cauchy不等式,(√a+√b+√c)² ≤ (a+b+c)(1+1+1) = 3,得√a+√b+√c ≤ √3.
同样由Cauchy不等式,((√a+√b)+(√b+√c)+(√c+√a))(1/(√a+√b)+1/(√b+√c)+1/(√c+√a)) ≥ (1+1+1)².
即得1/(√a+√b)+1/(√b+√c)+1/(√c+√a) ≥ 9/(2(√a+√b+√c)) ≥ 3√3/2.
同样由Cauchy不等式,((√a+√b)+(√b+√c)+(√c+√a))(1/(√a+√b)+1/(√b+√c)+1/(√c+√a)) ≥ (1+1+1)².
即得1/(√a+√b)+1/(√b+√c)+1/(√c+√a) ≥ 9/(2(√a+√b+√c)) ≥ 3√3/2.
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