<p>Determine the maximum value of \(z = 2x + 3y\), if the feasible region for an LPP is \(x + y \leq 4,\, x \geq 0,\, y \geq 0\).</p>
Step-by-Step Solution
Key Concept: The maximum value of a linear objective function over a convex feasible region always occurs at a vertex (corner point). Identify all vertices of the feasible region and evaluate the objective function at each to find the maximum.
<p><strong>Step 1:</strong> Identify the feasible region vertices by finding intersections of constraints:</p><ul><li>Intersection of x = 0 and y = 0: Point (0, 0)</li><li>Intersection of x + y = 4 and x = 0: Point (0, 4)</li><li>Intersection of x + y = 4 and y = 0: Point (4, 0)</li></ul><p><strong>Step 2:</strong> Evaluate z = 2x + 3y at each corner point:</p><ul><li>At (0, 0): z = 2(0) + 3(0) = 0</li><li>At (0, 4): z = 2(0) + 3(4) = 12</li><li>At (4, 0): z = 2(4) + 3(0) = 8</li></ul><p><strong>Step 3:</strong> Compare values: 0, 12, and 8. The maximum value is 12, occurring at the point (0, 4).</p><p>∴ Maximum value of z = <strong>12</strong></p>
Correct Answer: 12