Question

The feasible region represented by the constraints \(y-2x\leq 4,x+y\geq 5,x\leq 4,y\geq 2,x,y\geq 0\) is ...........

• A. a convex bounded region with 4 corner points
• B. an unbounded region
• C. a convex bounded region with 5 corner points
• D. no feasible region

Hint:

Find the corner points by pairing boundary lines.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Constraints: \(y - 2x \le 4\), \(x + y \ge 5\), \(x \le 4\), \(y \ge 2\), \(x, y \ge 0\). The feasible region is the intersection of all these half planes.

Step 2: Corner points:
\(y = 2\) with \(x + y = 5\): \((3, 2)\).
\(y = 2\) with \(x = 4\): \((4, 2)\).
\(x = 4\) with \(y = 2x + 4\): \((4, 12)\).
\(y = 2x + 4\) with \(x + y = 5\): \(x = \tfrac13\), \(y = \tfrac{14}{3}\): \(\left(\tfrac13, \tfrac{14}{3}\right)\).
Points on \(x = 0\) are excluded: there \(y \le 4\) by the first constraint, but \(x + y \ge 5\) needs \(y \ge 5\).

Step 3: Shape:
The region is enclosed by four lines: a quadrilateral with vertices \((3,2), (4,2), (4,12), (\tfrac13,\tfrac{14}{3})\). It is bounded and convex.

Step 4: Result:
A convex bounded region with 4 corner points, option (A).

Final Answer:
The region is a quadrilateral with four corners. \[ \boxed{\text{(A) }\text{convex bounded region with 4 corner points}} \]