Question

设abc为不全等的正数,abc等于1,求证1/a+1/b+1/c＞a^(1/2)+b^(1/2)+c^(1/2)

Answer 1

因为abc=1

所以,1/a+1/b+1/c

=abc/a+abc/+abc/c

=bc+ac+ab

=1/2(ab+bc+bc+ca+ca+ab)

＞1/2[2(ab^2c)^(1/2)+2(bc^2a)^(1/2)+2(a^2bc)^(1/2)] (因为abc不全相等,所以不能取等号)

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

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