a、b、c属于R+ 求证(b+c)/a+(a+c)/b+(a+b)/c≥6
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![a、b、c属于R+ 求证(b+c)/a+(a+c)/b+(a+b)/c≥6](/uploads/image/z/12502135-55-5.jpg?t=a%E3%80%81b%E3%80%81c%E5%B1%9E%E4%BA%8ER%2B+%E6%B1%82%E8%AF%81%28b%2Bc%29%2Fa%2B%28a%2Bc%29%2Fb%2B%28a%2Bb%29%2Fc%E2%89%A56)
a、b、c属于R+ 求证(b+c)/a+(a+c)/b+(a+b)/c≥6
a、b、c属于R+ 求证(b+c)/a+(a+c)/b+(a+b)/c≥6
a、b、c属于R+ 求证(b+c)/a+(a+c)/b+(a+b)/c≥6
(b+c)/a+(a+c)/b+(a+b)/c=b/a+c/a+a/b+c/b+a/c+b/c=(a/b+b/a)+(a/c+c/a)+(b/c+c/b) ∵a/b+b/a>=2√(a/b×b/a)=2(取等a/b=b/a,即a=b) a/c+c/a>=2√(a/c×c/a)=2(取等a/c=c/a,即a=c) b/c+c/b>=2√(b/c×c/b)=2(取等b/c=c/b,即b=c) ∴(b+c)/a+(a+c)/b+(a+b)/c=(a/b+b/a)+(a/c+c/a)+(b/c+c/b)>=2+2+2=6 即(b+c)/a+(a+c)/b+(a+b)/c>=6