Question

abc=1,求证1/a+1/b+1/c大于等于根号a+根号b+根号c

Answer 1

证明:

1/a+1/b+1/c=(ab+bc+ac)/abc

=ab+bc+ac

=(1/2)[(ab+bc)+(ab+ac)+(ac+bc)]

≥(1/2)[2(ab*bc)^(1/2)+2(ab+ac)^(1/2)+2(ac+bc)^(1/2)]

=(abc*b)^(1/2)+(abc*a)^(1/2)+(abc*c)^(1/2)

=b^(1/2)+a^(1/2)+c^(1/2)

得证.