已知a,b属于正实数,且a+b=1,求y=(a+1/a)(b+1/b)的最小值
问题描述:
已知a,b属于正实数,且a+b=1,求y=(a+1/a)(b+1/b)的最小值
答
(a+1/a)(b+1/b) =ab+b/a+a/b+1/(ab) =(a^2b^2+b^2+a^2+1)/(ab) =[a^2b^2+(a+b)^2-2ab+1]/(ab) =[a^2b^2+1-2ab+1]/(ab) =a^2b^2/ab-2ab/ab+2/ab=ab+2/ab-21=a+b>=2√(ab)所以√(ab)