Question

若正数x，y 满足x+3y=5xy，则3x+4y的最小值是

若正数x，y 满足x+3y=5xy，则3x+4y的最小值是

A. 24/5 B.28/5 C.5 D.6
回答为什么，不要只选答案。

Answer 1

x(1-5y)=-3y
x=3y/(5y-1)
3x+4y
=9y/(5y-1)+4y
=9(y-1/5+1/5)/(5y-1)+(4y-4/5)+4/5
=13/5+(9/5)/(5y-1)+4/5*(5y-1)
>=13/5+2√(9/5*4/5)
=13/5+12/5
=5
当且仅当(9/5)/(5y-1)=4/5*(5y-1)，即y=-1/10或1/2等号成立
所以当x=1/5 y=-1/10或x=1 y=1/2时，取到最小值5

Answer 2

3x+4y=k

y=(k-3x)/4
代入x+3y=5xy
x+3/4(k-3x)=5x(k-3x)/4
4x+3k-9x=5kx-15x²
15x²-(5k+5)x+3k=0
Δ=25(k+1)²-180k>=0
5(k+1)²-36k>=0

5k²+10k+5-36k>=0
5k²-26k+5>=0
(5k-1)(k-5)>=0
k<=1/5或k>=5
所以
最小值=5
选C

Answer 3

∵x+3y=5xy
∴(1/y)+(3/x)=5.
5(3x+4y)
=[(1/y)+(3/x)][4y+3x)≥(2+3)²=25
∴3x+4y≥5
∴(3x+4y)min=5

选择c