Minimize Z - x - 2 y subject to 2x - y - 3- x - 2y - 6 - x- y - 0- Show that the minimum of Z occurs at more than points-

2x+y≥ 3 #10; #10; #10; #10; #10; x #10; #10; #10; 0 #10; #10; #10; 1 #10; #10; #10; #10; #10; y #10 ...

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Byju's Answer

Standard XII

Mathematics

Local Maxima

Minimize Z ...

Question

Minimize $Z = x + 2 y$ subject to $2 x + y \geq 3, x + 2 y \geq 6, x, y \geq 0$ . Show that the minimum of $Z$ occurs at more than points.

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Solution

$2 x + y \geq 3$

x

0

1

y

3

1

$x + 2 y \geq 6$

x

0

6

y

3

0

Corner points

Value of Z

(0,3)

6

(6,0)

6

Since, the region that is feasible is unbounded,

Hence 6 may or may not be the minimum value of Z.

We need the graph of inequality

$x + 2 y < 6$

x

0

6

y

3

0

There is no common point between the feasible region and inequality.

Z will be minimum on the line $x + 2 y = 6$ .

x

y

Points

Value of Z

2

2

(2,2)

2+2(2)=6

4

1

(4,1)

4+2(1)=6

-2

4

(-2,4)

-2+2(4)=6

Z is minimum at all the points joining (0,3) and (6,0)

Therefore Z will be minimum on $x + 2 y = 6$

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x	y	Points	Value of Z
2	2	(2,2)	2+2(2)=6
4	1	(4,1)	4+2(1)=6
-2	4	(-2,4)	-2+2(4)=6

Minimize Z=x+2y subject to 2x+y≥3,x+2y≥6,x,y≥0. Show that the minimum of Z occurs at more than points.

Minimize $Z = x + 2 y$ subject to $2 x + y \geq 3, x + 2 y \geq 6, x, y \geq 0$ . Show that the minimum of $Z$ occurs at more than points.