已知a>0,b>0,c>0,且a+b+c=1,求证1⼀a+1⼀b+1⼀c≥9

已知a>0,b>0,c>0,且a+b+c=1,求证1/a+1/b+1/c≥9 详解10分
2025-03-11 06:16:30
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回答1:

1/a+1/b+1/c =(a+b+c)/a+(a+b+c)/b+(a+b+c)/c =1+(b+c)/a+1(a+c)/b+1(a+b)/c =3+b/c+c/b+a/c+c/a+a/b+b/a (由于b/a+a/b>=2,c/a+a/c>=2,c/b+b/c>=2) >=3+2+2+2 =9