Consider the feasible region determined by the constraints \(x + y \le 8\), \(x \le 6\), \(y \le 5\), \(x \ge 0\), \(y \ge 0\). Match each objective function in List-I with where its maximum on this region occurs, in List-II.
| List-I | List-II | ||
|---|---|---|---|
| Objective function | Maximum occurs | ||
| (A) | \(Z = 2x + y\) | (I) | At every point of the segment joining (3, 5) and (0, 5) |
| (B) | \(Z = x + 2y\) | (II) | Only at (6, 2) |
| (C) | \(Z = 3x + 3y\) | (III) | At every point of the segment joining (6, 2) and (3, 5) |
| (D) | \(Z = 4y\) | (IV) | Only at (3, 5) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(II), (B)-(IV), (C)-(III), (D)-(I)
The line \(x + y = 8\) meets \(x = 6\) at (6, 2) and \(y = 5\) at (3, 5). So the corner points are (0, 0), (6, 0), (6, 2), (3, 5) and (0, 5). If the largest value occurs at two adjacent corners, every point of the edge between them is also a maximum point.
(A) \(Z = 2x + y\)
Corner values: \(0, \; 12, \; 14, \; 11, \; 5\)
This matches (II) Only at (6, 2).
(B) \(Z = x + 2y\)
Corner values: \(0, \; 6, \; 10, \; 13, \; 10\)
This matches (IV) Only at (3, 5).
(C) \(Z = 3x + 3y\)
Corner values: \(0, \; 18, \; 24, \; 24, \; 15\)
This matches (III): the maximum 24 occurs at (6, 2) and (3, 5), so all along the edge joining them.
(D) \(Z = 4y\)
Corner values: \(0, \; 0, \; 8, \; 20, \; 20\)
This matches (I): the maximum 20 occurs at (3, 5) and (0, 5), so all along the edge joining them.