Consider the feasible region determined by the constraints \(x + 2y \le 12\), \(x \le 8\), \(y \le 5\), \(x \ge 0\), \(y \ge 0\), and the following statements:
(A) The maximum of \(Z = x + 2y\) on this region occurs at exactly one point.
(B) The feasible region has 5 corner points.
(C) The maximum value of \(Z = 2x + y\) on this region is 18.
(D) \((2, 6)\) is a corner point of the feasible region.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B) and (C) only
The line \(x + 2y = 12\) meets \(x = 8\) where
\(8 + 2y = 12\)
\(y = 2\)
It meets \(y = 5\) where
\(x + 10 = 12\)
\(x = 2\)
So the bounded region has the corner points (0, 0), (8, 0), (8, 2), (2, 5) and (0, 5).
Values of \(Z = x + 2y\):
\(0, \; 8, \; 12, \; 12, \; 10\)
Values of \(Z = 2x + y\):
\(0, \; 16, \; 18, \; 9, \; 5\)
So the region has 5 corner points, \(x + 2y\) has maximum 12 at both (8, 2) and (2, 5), and \(2x + y\) has maximum 18 at (8, 2).
Checking the statements
(A) The maximum of \(Z = x + 2y\) on this region occurs at exactly one point. — Incorrect. The value 12 occurs at (8, 2) and (2, 5), so along the whole edge between them.
(B) The feasible region has 5 corner points. — Correct. (0, 0), (8, 0), (8, 2), (2, 5), (0, 5).
(C) The maximum value of \(Z = 2x + y\) on this region is 18. — Correct. \(Z(8, 2) = 18\) is the largest corner value.
(D) \((2, 6)\) is a corner point of the feasible region. — Incorrect. \(y = 6\) breaks \(y \le 5\), so the point is not even feasible.