<p>The feasible region for an LPP is shown in the following figure. Let \(z = 3x - 4y\) be the objective function. Maximum value of \(z\) is. (Corner points: \((0,4)\), \((6,0)\), \((12,6)\))</p>
Step-by-Step Solution
Key Concept: For a linear objective function over a convex feasible region, the maximum (or minimum) always occurs at a corner point of the feasible region. Evaluate z at all corner points and identify the largest value.
<p><strong>Step 1:</strong> The objective function is z = 3x - 4y. For a linear program, the optimal value occurs at a corner point of the feasible region.</p><p><strong>Step 2:</strong> Evaluate z at each corner point:</p><ul><li>At (0, 4): z = 3(0) - 4(4) = 0 - 16 = <strong>-16</strong></li><li>At (6, 0): z = 3(6) - 4(0) = 18 - 0 = <strong>18</strong></li><li>At (12, 6): z = 3(12) - 4(6) = 36 - 24 = <strong>12</strong></li></ul><p><strong>Step 3:</strong> Compare all values: -16, 18, and 12. The maximum value is 18, occurring at the point (6, 0).</p><p><strong>Note:</strong> If the question asks for the maximum value and the answer given is 0, there may be a different configuration. However, based on standard LPP methodology with the given objective function and corner points, the maximum is <strong>18</strong>. If the answer is indeed 0, verify the problem statement (the objective function or feasible region definition may differ).</p>
Correct Answer: 0