Consider the feasible region determined by the constraints \(x \le 6\), \(y \le 6\), \(x + y \ge 2\), \(x + y \le 10\), \(x \ge 0\), \(y \ge 0\), and the following statements:
(A) The feasible region has 6 corner points.
(B) The maximum value of \(Z = 3x + 2y\) on this region is 24.
(C) The minimum value of \(Z = 3x + 2y\) on this region is 4.
(D) The maximum of \(Z = x + y\) on this region occurs at exactly two points.
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 line \(x + y = 10\) meets \(x = 6\) at (6, 4) and \(y = 6\) at (4, 6). The line \(x + y = 2\) meets the axes at (2, 0) and (0, 2).
So the corner points are (2, 0), (6, 0), (6, 4), (4, 6), (0, 6) and (0, 2).
Values of \(Z = 3x + 2y\):
\(6, \; 18, \; 26, \; 24, \; 12, \; 4\)
Values of \(Z = x + y\):
\(2, \; 6, \; 10, \; 10, \; 6, \; 2\)
So the region has 6 corners, \(3x + 2y\) has maximum 26 and minimum 4, and \(x + y\) has maximum 10 along the whole edge from (6, 4) to (4, 6).
Checking the statements
(A) The feasible region has 6 corner points. — Correct. (2, 0), (6, 0), (6, 4), (4, 6), (0, 6), (0, 2).
(B) The maximum value of \(Z = 3x + 2y\) on this region is 24. — Incorrect. \(Z(6, 4) = 26\) is larger.
(C) The minimum value of \(Z = 3x + 2y\) on this region is 4. — Correct. \(Z(0, 2) = 4\) is the smallest corner value.
(D) The maximum of \(Z = x + y\) on this region occurs at exactly two points. — Incorrect. Every point of the edge from (6, 4) to (4, 6) gives 10, so there are infinitely many.