Question

The number of corner points of the feasible region determined by the constraints \( x \geq 0, y \geq 0, x + y \geq 4 \) is:

• A. 0
• B. 1
• C. 2
• D. 3
• E.

Hint:

For linear programming problems, the corner points of the feasible region are determined by the intersection of the constraints. Always consider the boundary conditions of the inequalities.

Answer 1

The Correct Option is C

The constraints are: \[ x \geq 0, \quad y \geq 0, \quad x + y \geq 4. \] 1. Interpretation of Constraints: - \( x \geq 0 \): The region lies to the right of the \( y \)-axis. - \( y \geq 0 \): The region lies above the \( x \)-axis. - \( x + y \geq 4 \): The region is above or on the line \( x + y = 4 \). 2. Feasible Region: The feasible region is determined by the intersection of these constraints in the first quadrant. The line \( x + y = 4 \) intersects the axes at: - \( x = 4, y = 0 \) (on the \( x \)-axis), - \( x = 0, y = 4 \) (on the \( y \)-axis). Since \( x \geq 0 \) and \( y \geq 0 \), these points form the boundary of the feasible region. The feasible region is unbounded but includes two corner points at: - \( (4, 0) \), - \( (0, 4) \). 3. Number of Corner Points: The feasible region has exactly two corner points. Hence, the correct answer is (C) 2.