For the objective function \(Z = 2x + y\) subject to the constraints \(x + 2y \ge 4\), \(3x + 2y \le 12\), \(x \ge 0\), \(y \ge 0\), the value of (maximum value of Z) − (minimum value of Z) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 6
Both lines \(x + 2y = 4\) and \(3x + 2y = 12\) cross the x-axis at the same point (4, 0). On the y-axis they give (0, 2) and (0, 6).
The feasible region lies above the first line and below the second, so it is the triangle with corners (0, 2), (0, 6) and (4, 0). It is bounded.
\(Z(0, 2) = 2\)
\(Z(0, 6) = 6\)
\(Z(4, 0) = 8\)
The maximum is 8, at (4, 0), and the minimum is 2, at (0, 2).
\(8 - 2 = 6\)
Why the other options are incorrect
Option (1) 8 — Incorrect. This is the maximum of Z alone, not the difference.
Option (2) 4 — Incorrect. This is 6 − 2, taking Z(0, 6) = 6 as the maximum; Z(4, 0) = 8 is larger.
Option (3) 10 — Incorrect. This is 8 + 2, the sum of the maximum and the minimum, not their difference.