If the corner points of the bounded feasible region for a Linear Programming Problem (LPP) are A(0, 2), B(3, 0), C(2, 3) and D(3, 1), then the maximum value of the objective function \(Z = 4x + 2y\) occurs at
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → every point on the line segment joining the points (2, 3) and (3, 1)
By the corner point method, Z is worked out at each corner:
A(0, 2): Z = 0 + 4 = 4
B(3, 0): Z = 12 + 0 = 12
C(2, 3): Z = 8 + 6 = 14
D(3, 1): Z = 12 + 2 = 14
The maximum 14 occurs at two corners, C and D. When the maximum occurs at two adjacent corners, it occurs at every point of the segment joining them. Here the segment CD lies on the line \(4x + 2y = 14\), so Z = 14 all along it.
Why the other options are incorrect
Option (1) (0, 2) only — Incorrect. Z = 4 at (0, 2) is the minimum, not the maximum.
Option (2) the mid-point of the line segment joining the points (2, 3) and (3, 1) only — Incorrect. The mid-point (2.5, 2) does give Z = 14, but so does every other point of the segment, so “only” is wrong.
Option (3) (2, 3) and (3, 1) only — Incorrect. The two corners are not the only points; all the points between them also give Z = 14.