Question

已知x,y属于(0,正无穷),且满足xy=x+y+3,求xy的最小值.

Answer 1

设t=xy

则：x=t/y

xy=x+y+3

t=t/y+y+3

y^2+(3-t)y+t=0

△=(3-t)^2-4t

=9-10t+t^2

=(t-1)(t-9)≥0

t≥9,或,t≤1

因为x,y大于0,所以,y1+y2=t-3>0

t>3

所以,t≥9

即：xy的最小值为9