4.2　简单线性规划

第1课时　求线性目标函数的最值

课时过关·能力提升

1.若变量x,y满足约束条件{■(x+y≤8"," @2y"-" x≤4"," @x≥0"," @y≥0"," )┤且z=5y-x的最大值为a,最小值为b,则a-b的值是(　　)

A.48 B.30 C.24 D.16

解析:画出可行域,如图所示.

　　联立{■(x+y=8"," @2y"-" x=4"," )┤解得{■(x=4"," @y=4"." )┤即点A坐标为(4,4).

　　故zmax=5×4-4=16,zmin=0-8=-8,即a=16,b=-8,因此a-b=24.故选C.

答案:C

2.若变量x,y满足约束条件{■(x+y≥0"," @2x"-" y≥0"," @x≤4"," )┤则z=2x+y取最大值时的最优解为(　　)

A.(4,8) B.(4,-4) C.16 D.4

解析:在直角坐标系内画出不等式组表示的平面区域,如图中阴影部分所示,当直线y=-2x+z经过点A(4,8)时,纵截距z取得最大值16,因此z=2x+y取最大值时的最优解为(4,8).

答案:A

3.若变量x,y满足{■(x+y"-" 1≥0"," @x+3y"-" 6≤0"," @x"-" y"-" 2≥0"," )┤则z=x-2y的最小值为(　　)

A.1 B.5/2 C.3 D.3/2

解析:画出不等式组表示的平面区域,如图中阴影部分所示.

　　当直线y=1/2x-1/2z经过点A(3,1)时,在y轴上的截距-1/2z达到最大值,此时z取得最小值1.

答案:A

4.设实数x,y满足不等式组{■(x+2y"-" 5>0"," @2x+y"-" 7>0"," @x≥0"," y≥0"," )┤若x,y为整数,则z=3x+4y的最小值是(　　)