x+3y=5xy,(x,y>0)求3x+y最小值

x+3y=5xy,(x,y>0)求3x+y最小值
x+3y=5xy,(x,y>0)求3x+y最小值

x+3y=5xy,(x,y>0)求3x+y最小值
x+3y=5xy
x(5y-1) =3y
x = 3y/(5y-1)
S = 3x +y
=9y/(5y-1) +y
= (9/5)+ (9/5)[1/(5y-1)] +y
S' = -(9)[1/(5y-1)^2] +1 =0
(5y-1)^2-9=0
25y^2-10y-8=0
(5y+2)(5y-4)=0
y= 4/5
S'' = (90)[1/(5y-1)^3]
S''(4/5)>0 (min)
S = 9y/(5y-1) +y
minS = S(4/5)
S(4/5) = (36/5)/3 + 4/5
= 12/5 +4/5
=16/5

∵x+3y=5xy,且x,y＞0
∴两边同除以xy,可得
(1/y)+(3/x)=5
∴由柯西不等式可得：
5(3x+y)
=[(3/x)+(1/y)]×(3x+y)≥(3+1)²=16
∴3x+4y≥4
∴(3x+4y)min=4

解:因为x＋3y=5xy,所以1/(5y)＋3/(5x)=1,所以3x＋y=(3x＋y)[1/(5y)＋3/(5x)]=(3x)/(5y)＋1/5＋9/5＋(3y)/(5x)=(3x)/(5y)＋(3y)/(5x)＋1/5＋9/5=(3y)/(5x)＋(3x)/(5y)＋2=<根号[(3y)/(5x)*(3x)/(5y)]＋2=3/5＋2=13/5,所以3x＋y的最小值为13/5。

答案最小值
16/5