Question

设x>0,y>0,且x+y=1,求证:(x+1/x)(y+1/y)>=25/4

Answer 1

设

x = cos^2t

y = sin^2t

(x+1/x)(y+1/y)

= sin^2t*cos^2t + 1/sin^2tcos^2t + sin^2t/cos^2t +cos^2t/sin^2t

= (2 - 2sin^2tcos^2t + sin^4tcos^4t)/sin^2tcos^2t

= sin^2tcos^2t + 2/sin^2tcos^2t - 2

= sin^2(2t)/4 + 8/sin^2(2t) - 2

0 < sin^2(2t) = 25/4