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>=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