For the objective function \(Z = 2x + 3y\) subject to the constraints \(2x + y \ge 6\), \(x + 2y \le 12\), \(x \le 6\), \(x \ge 0\), \(y \ge 0\), the points at which Z attains its maximum and its minimum value are respectively
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \((6, 3)\) and \((3, 0)\)
Both lines \(2x + y = 6\) and \(x + 2y = 12\) pass through (0, 6). The line \(2x + y = 6\) meets the x-axis at (3, 0), and \(x + 2y = 12\) meets \(x = 6\) at (6, 3).
So the region is bounded, with corner points (3, 0), (6, 0), (6, 3) and (0, 6). Values of Z:
\(Z(3, 0) = 6\)
\(Z(6, 0) = 12\)
\(Z(6, 3) = 12 + 9 = 21\)
\(Z(0, 6) = 18\)
The maximum 21 is at (6, 3) and the minimum 6 is at (3, 0).
Why the other options are incorrect
Option (1) \((0, 6)\) and \((3, 0)\) — Incorrect. \(Z(0, 6) = 18\) is less than \(Z(6, 3) = 21\).
Option (2) \((6, 3)\) and \((6, 0)\) — Incorrect. \(Z(6, 0) = 12\) is more than \(Z(3, 0) = 6\).
Option (3) \((0, 6)\) and \((0, 0)\) — Incorrect. The origin breaks \(2x + y \ge 6\), so it is not feasible.