Question

已知/x/=≤1,/y/≤1,设M＝/x+y/+/y+1/+/2y-x-4/,求M最大值和最小ŀ

Answer 1

解：
∵|y|≤1
∴-1
≤
y
≤1
∴
y
+1≥0，
∴
|y+1|
=
y
+1
|2y-x|
≤
|2y|+|x|
≤
2
+1
=3
∴2y-x-4
＜0
∴|2y-x-4|
=
-(2y-x-4)
(1)若x+y≥0，则
m
=|x+y|+|y+1|+|2y-x-4|
=
(x+y)
+(y+1)-(2y-x-4)
=
2x
+5
∵-1≤x≤1
∴3≤m≤7．
(2)若x+y＜0，则
m
=
-(x+y)
+(y+1)
-(2y-x-4)
=
-2y
+5
∵-1
≤
y
≤1
∴3
≤
m
≤
7，
综上得3≤
m
≤7．
m的最大值是7，(x
=
1,
y
=
-1时)
最小值是3.（x
=
-1,y
=
1时)