The point at which \(Z = 2x - 3y\) attains its minimum value, subject to the constraints \(x + y \le 5\), \(x \le 3\), \(y \le 4\), \(x \ge 0\), \(y \ge 0\), is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (0, 4)
The region is bounded, so the minimum of Z is at a corner point. Along with (0, 0), (3, 0) and (0, 4), the line x + y = 5 cuts two more corners:
with x = 3: \(y = 5 - 3 = 2\), giving (3, 2)
with y = 4: \(x = 5 - 4 = 1\), giving (1, 4)
Values of Z = 2x − 3y at the corners:
\(Z(0, 0) = 0\)
\(Z(3, 0) = 6\)
\(Z(3, 2) = 6 - 6 = 0\)
\(Z(1, 4) = 2 - 12 = -10\)
\(Z(0, 4) = -12\)
The minimum is −12, at (0, 4).
Why the other options are incorrect
Option (1) (1, 4) — Incorrect. There Z = −10, which is larger than −12.
Option (3) (0, 0) — Incorrect. There Z = 0, well above −12.
Option (4) (3, 0) — Incorrect. There Z = 6, the maximum of Z, not the minimum.