Question

变量xy,满足线性约束条件{3x+y-2≤0,y-x≤2,y≥-x-1,则目标函数z=kx-y仅在点(0,2)

变量xy,满足线性约束条件{3x+y-2≤0,y-x≤2,y≥-x-1,则目标函数z=kx-y仅在点(0,2)取得最小值，求k的取值范围
要过程

Answer 1

3x+y-2≤0 (1')
y-x≤2 (2')
y≥-x-1 (3')

3x+y-2=0 (1)
y-x=2 (2)
y=-x-1 (3)

z=kx-y
min z = z(0,2) = -2

case 1: from (1) and (2)
x=0
y=2

(x,y)=(0,2) 满足(3')
z=kx-y
z(0,2)=-2
ie min case

case 2: from (1) and (3)
(1)-(3)
2x=3
x=3/2
from (1) y=-5/2
(x,y)=(3/2,-5/2) 满足(2')
z=kx-y
z(3/2,-5/2)=3k/2+5/2 >中咐拦-2
k > -3

case 3:from (2) and (3)
(2)+(3)
2y=1
y=1/卖胡2
from (2), x=-3/2

(x,y)=(-3/2,1/2) 满足(2')
z=kx-y
z(-3/简裤2,1/2)=-3k/2-1/2 >-2
k < 1

case 2 and case 3
ie
-3<k<1