Question

已知a,b,c属于R+,且a+b+c=1.求证:(1/a-1)(1/b-1)(1/c-1)≥8

Answer 1

∵a,b,c为正实数,且a+b+c=1,∴1/a-1=(a+b+c)/a-1=(b+c)/a≥(2√bc)/a>0；1/b-1=(a+b+c)/b-1=(c+a)/b≥(2√ca)/b>0；1/c-1=(a+b+c)/c-1=(a+b)/c≥(2√ab)/c>0,三式相乘,得(1/a-1)(1/b-1)(1/c-1)≥[(2√bc)/a] [(2√ca...