用综合法证明,设a>0,b>0且a+b=1,则(a+1/a)^2+(b+1/b)^2大于等于25/2

Question

用综合法证明,设a>0,b>0且a+b=1,则(a +1/a)^2+(b+ 1/b)^2大于等于25/2

Answer 1

(a+1/a)^2+(b+1/b)^2
≥[(a+1/a)+(b+1/b)]^2/2
=(a+(a+b)/a+b+(a+b)/b]^2/2
=(a+b+1+b/a+1+a/b)^2/2
=(3+b/a+a/b)^2/2
≥(3+2)^2/2
=25/2