Question

Show that the maximum of $Z$ occurs at more than two points.
Minimise and Maximise $Z=5x+10y$ subject to $x+2y\le 120,x+y\ge 60,x-2y\ge 60,x-2y\ge 0,x,y\ge 0$.

Answer 1

Objective function : $Z=5x+10y$

We have to minimize and maximize Z on constraints

$x+2y\le 120$

$x+y\ge 60$

$x-2y\ge 60$

$x-2y\ge 0$

$x\ge 0,y\ge 0$

After plotting all the constraints on coordinate plane, we get the feasible region as shown in the image.

Corner points	Value of $Z=5x+10y$
(60,0)	300 (Minimum)
(90,15)	600 (Maximum)
(120,0)	600 (Maximum)

Here maximum of $Z=600$

Plotting the line $Z=600\Rightarrow 5x+10y=600$

Since this line coincide with line $x+2y=120$ , Hence all the points lying on line $x+2y=120$ in feasible region will give maximum value.

So, there are more than 2 values where maximum of Z will occur.