A linear programming problem is as follows:
Minimize \(z = 2x + 3y\)
Subject to the constraints \(x \ge 3, x \le 9, y \ge 0, x-y \ge 0, x + y \le 14\). The feasible region has 5 corner points including
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (7, 7) and (3, 3)
The feasible region lies between the lines x = 3 and x = 9, above y = 0, on or below y = x, and on or below x + y = 14. Its corner points are where these boundary lines meet:
\(x = 3,\ y = 0:\quad (3, 0)\)
\(x = 9,\ y = 0:\quad (9, 0)\)
\(x = 9,\ x + y = 14:\quad (9, 5)\)
\(y = x,\ x + y = 14:\quad (7, 7)\)
\(x = 3,\ y = x:\quad (3, 3)\)
Each of these satisfies all the constraints, so the five corner points are (3, 0), (9, 0), (9, 5), (7, 7) and (3, 3).
Option (1) (0, 0) and (9, 5) — Incorrect. (0, 0) violates x ≥ 3.
Option (2) (14, 0) and (9, 0) — Incorrect. (14, 0) violates x ≤ 9.
Option (3) (7, 7) and (3, 3) — Correct. (7, 7) is where y = x meets x + y = 14, and (3, 3) is where x = 3 meets y = x; both are corner points.
Option (4) (3, 6) and (9, 5) — Incorrect. (3, 6) violates x − y ≥ 0, since 3 − 6 < 0.