Question

已知a>0,b>0且a+b=1,求证：(a+1/a)^2+(b+1/b)^2>25/4

Answer 1

(a+1/a)^2+(b+1/b)^2

=a^2+1/a^2+2 +b^2+1/b^2+2

=(a^2+b^2) + (1/a^2+1/b^2) +4

>=0.5*(a+b)^2 +0.5*(1/a +1/b)^2 +4

=0.5+ 0.5*(1/a +1/b)^2+4

=4.5+0.5*(1/a+1/b)^2

因为ab=4；

(1/a +1/b)^2>=16

所以(a+1/a)^2+(b+1/b)^2>=4.5+0.5*16=25/2