Question

The maximum value of $z=2x+5y$ subject to the constraints $2x+5y\le 10,x+2y\ge 1,x-y\le 4,x\ge 0,y\ge 0$, occurs at

• A. exactly one point.
• B. exactly two points
• C. infinitely many points.
• D. None of these.

Answer 1

Answer: (C)

We find that the feasible region is on the same side of the line $2x+5y=10$ as the origin, on the same side of the line $x-y=4$ as the origin and on the opposite side of the line $x+2y=1$ from the origin. Moreover, the lines meet the coordinate axes at $(5,0),(0,2);(1,0),(0,1/2)$ and $(4,0)$.
The lines $x-y=4$ and $2x+5y=10$ intersect at $\left(\frac{30}{7},\frac{2}{7}\right)$
The values of the objective function at the vertices of the pentagon are:
(a) $0+\frac{5}{2}=\frac{5}{2}$
(b) $2+0=2$
(c) $8+0=8$
(d) $\frac{60}{7}+\frac{10}{7}=10$
(e) $0+10=10$
The maximum value 10 occurs at the points $D\left(\frac{30}{7},\frac{2}{7}\right)$ and $E(0,2),$ Since D and E are adjacent vertices, the objective function has the same maximum value 10 at all the points on the lines DE.