The minimum value of \(Z = x + 2y\) subject to the constraints \(x + y \ge 4\), \(x - y \le 0\), \(y \le 6\), \(x \ge 0\), \(y \ge 0\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 6
The constraint \(x - y \le 0\) means \(y \ge x\): the region lies above the line y = x. It also lies above x + y = 4, below y = 6 and to the right of the y-axis. It is bounded.
Its corner points:
y = x with x + y = 4: (2, 2)
y = x with y = 6: (6, 6)
x = 0 with y = 6: (0, 6)
x = 0 with x + y = 4: (0, 4)
Values of Z = x + 2y:
\(Z(2, 2) = 2 + 4 = 6\)
\(Z(6, 6) = 6 + 12 = 18\)
\(Z(0, 6) = 12\)
\(Z(0, 4) = 8\)
The minimum is 6, at (2, 2).
Why the other options are incorrect
Option (1) 8 — Incorrect. This is Z at the corner (0, 4), which is larger than 6.
Option (2) 4 — Incorrect. This is Z at (4, 0), which is not feasible: there x − y = 4 > 0.
Option (4) 0 — Incorrect. This is Z at (0, 0), which is not feasible: there x + y = 0 < 4.