用分析法证明:若a>0,则根号下(a^2+1/a^2)-根号2>a+1/a-2

问题描述:

用分析法证明:若a>0,则根号下(a^2+1/a^2)-根号2>a+1/a-2

欲证√[a^2+1/(a^2)]-√2>a+1/a-2,则证{√[a^2+1/(a^2)]-√2}/(a+1/a-2)>1;分子有理化,得:{(a^2+1/a^2-2)}/{(a+1/a-2)(√[a^2+1/(a^2)]+√2)}=>{(a-1/a)^2}/{(a+1/a-2)(√[a^2+1/(a^2)]+√2)}=>{(√a+1/√a)^...