x,y都大于0,1/x+2/y=2,求xy最小值
问题描述：
x,y都大于0,1/x+2/y=2,求xy最小值
答
1/x=2-2/y =2(y-1)/y
x=y/(2y-2)
xy=y^2/(2y-2)
=1/2*(y^2-2y+1+2y-1)/(y-1)
=1/2 (y-1) +1/2*(2y-1)/(y-1)
=1/2 *(y-1)+1/2*(2y-2+1)/(y-1)
=1/2*(y-1)+1+1/2*1/(y-1)
>=1+1/2*2根号((y-1)*1/(y-1)) 等号的条件是y-1=1/(y-1) y-1=1 y=2
=2 (注意到x,y>0 则y>1 若y2 则1/x+2/y>2）
所以最小值为2
答
1/x+2/y>=2√(2/xy)
2>=2√(2/xy)
1>=√(2/xy)
1>=2/xy
xy>=2
xy最小值2