The corner points of the feasible region are (0,0), (2,4), (0,5), and (4,0).
The objective function is z=ax+by where a,b>0.
The maximum value occurs at both (2,4) and (4,0).
In Linear Programming, the maximum value of z=ax+by always occurs at a corner point of the feasible region.
When the maximum occurs at two corner points simultaneously, both points give the same maximum value of z, and the line z=ax+by is parallel to the edge connecting these two points.
Since the maximum occurs at both (2,4) and (4,0), the value of z at these points must be equal.
At point (2,4):
z=a(2)+b(4)
z=2a+4b
At point (4,0):
z=a(4)+b(0)
z=4a
Setting the values equal:
2a+4b=4a
4b=4a−2a
4b=2a
2b=a
Therefore a=2b.