Question

已知x,y是正数,且xy+x+y=1,则xy的最大值与x+y的最小值分别为

Answer 1

x,y是正数
xy+x+y=1
xy+x+y+1=2
x(y+1)+y+1=2
(y+1)(x+1)=2
x>0,y>0
x+1>0,y+1>0
x+1+y+1>=2√[(x+1)(y+1)]=2√2
x+y>=2(√2-1)
x+y的最小值为：2(√2-1)

xy=1-(x+y)<=1-2(√2-1)=3-2√2
xy的最大值为：3-2√2

Answer 2

xy+x+y=1
x(1+y)=1-y
x=(1-y)/(1+y)
xy=(1-y)y/(1+y)
=(-y^2+y)/(1+y)
=(-y^2-y+2y+2-2)/(1+y)
=-y+2-2/(1+y)
=-y-1-2/(1+y)+3
=-(y+1)-2/(1+y)+3
当(y+1)=2/(y+1)时，xy有最大值。
此时(y+1)^2=2 y+1=√2 y=√2-1
(xy)max=3-2√2

x+y=(1-y)/(1+y)+y
=(1-y+y+y^2)/(1+y)
=(y^2+1)/(1+y)
=(y^2+y-y-1+2)/(1+y)
=y-1+2/(1+y)
=(y+1)+2/(1+y)-2
(y+1)=2/(1+y)时,x+y有最小值。
此时y=√2-1
(x+y)min=2=2√2-2