Question

记A的平方为B 求证：根号（B+1/B) - 根号2 》 A+1/A - 2

Answer 1

注意到（A+1/A)^2=a^2 + 1/a^2 + 2

设 C=A+1/A

C0时

则 左边 = (C^2 - 2)^(1/2) - 2^(1/2) = (C^2 - 4)/((C^2-2)^(1/2) + 2^(1/2))

= (C - 2)(C + 2)/((C^2-2)^(1/2) + 2^(1/2))>=(C - 2)(C + 2)/(C + 2^(1/2)) > C - 2 = 右边

其中“>=”是由于|C| >= 2.