Consider the linear programming problem with objective function \(Z = 3x + 2y\) and constraints \(x + y \le 6\), \(2x + y \ge 4\), \(x \ge 0\), \(y \ge 0\).
(A) The feasible region is bounded
(B) The feasible region has exactly 3 corner points
(C) The maximum value of Z is 18
(D) The minimum value of Z is 8
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
The region lies below the line x + y = 6 and on the far side of the line 2x + y = 4 from the origin, in the first quadrant.
x + y = 6 meets the axes at (6, 0) and (0, 6)
2x + y = 4 meets the axes at (2, 0) and (0, 4)
The two lines do not meet in the first quadrant (solving gives x = −2), so the region is the quadrilateral with corner points (2, 0), (6, 0), (0, 6) and (0, 4).
\(Z(2, 0) = 6\)
\(Z(6, 0) = 18\)
\(Z(0, 6) = 12\)
\(Z(0, 4) = 8\)
So the region is bounded with 4 corner points, the maximum of Z is 18 and the minimum of Z is 6.
Checking the statements
(A) The feasible region is bounded — Correct. It lies inside the triangle cut off by x + y = 6.
(B) The feasible region has exactly 3 corner points — Incorrect. It has 4: (2, 0), (6, 0), (0, 6), (0, 4).
(C) The maximum value of Z is 18 — Correct. \(Z(6, 0) = 18\) is the largest corner value.
(D) The minimum value of Z is 8 — Incorrect. \(Z(2, 0) = 6\) is smaller; 8 is Z at (0, 4).