Question

若正实数x,y满足2x+y+6=xy,x+y的最小值是

Answer 1

2x+y+6=xy

y=(2x+6)/(x-1) >0

∴ x>1

x+y

=x+(2x+6)/(x-1)

=x+[(2x-2)+8]/(x-1)

=x+2+8/(x-1)

=(x-1)+8/(x-1)+3

≥2√8 +3

=4√2 +3

当且仅当 x-1=8/(x-1),即 x=2√2+1时等号成立

所以 x+y的最小值是4√2 +3