Consider the feasible region determined by the constraints \(y \ge 1\), \(x \ge 2y\), \(x + y \le 9\), and the following statements:
(A) The feasible region is a triangle.
(B) The origin is a feasible solution.
(C) The maximum value of \(Z = x + 4y\) on this region is 18.
(D) The minimum value of \(Z = x + 4y\) on this region is 0.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
The region lies above \(y = 1\), to the right of the line \(x = 2y\) and below \(x + y = 9\). The three lines meet in pairs at
\(y = 1\) and \(x = 2y\): (2, 1)
\(y = 1\) and \(x + y = 9\): (8, 1)
\(x = 2y\) and \(x + y = 9\): \(3y = 9\), so (6, 3)
Values of \(Z = x + 4y\) at these corners:
\(Z(2, 1) = 6\), \(Z(8, 1) = 12\), \(Z(6, 3) = 18\)
So the feasible region is the triangle with corners (2, 1), (8, 1), (6, 3); on it Z has maximum 18 and minimum 6.
Checking the statements
(A) The feasible region is a triangle. — Correct. Its corners are (2, 1), (8, 1) and (6, 3).
(B) The origin is a feasible solution. — Incorrect. At (0, 0), \(y = 0 \lt 1\).
(C) The maximum value of \(Z = x + 4y\) on this region is 18. — Correct. Z = 18 at (6, 3), the largest corner value.
(D) The minimum value of \(Z = x + 4y\) on this region is 0. — Incorrect. The minimum is 6, at (2, 1).