Question

已知abc都是正实数,求证a2 b2 c2>=1/3(a b c)2>=ab bc ac

Answer 1

证明：由abc=1带入
有(1/a^2)+(1/b^2)+(1/c^2)=abc/a^2+abc/b^2+abc/c^2=bc/a+ac/b+ab/c
=1/2[(bc/a)+(ac/b)]+1/2[(bc/a)+(ab/c)]+1/2[(ac/b)+(ab/c)]
再根据基本不等式有
[(bc/a)+(ac/b)]>=2根号下[(bc/a)*(ac/b)]=2c
[(bc/a)+(ab/c)]>=2根号下[(bc/a)*(ab/c)]=2b
[(ac/b)+(ab/c)]>=2根号下[(ac/b)*(ab/c)]=2a
再把上面的3个式子相加得到
(1/a2)+(1/b2)+(1/c2)>=a+b+c