x+3y=5xy,(x,y>0)求3x+y最小值
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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(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
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