Relations & Functions
Linear Programming - Minimum of objective function
Grade None
Question:
<p>Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let \(z = 4x + 6y\) be the objective function. The minimum value of <em>z</em> occurs at</p>
<p>(0, 2) only</p>
<p>(3, 0) only</p>
<p>the midpoint of the line segment joining the points (0, 2) and (3, 0) only</p>
<p>any point on the line segment joining the points (0, 2) and (3, 0).</p>
Step-by-Step Solution
Key Concept: In Linear Programming Problems, the optimal value of a linear objective function over a convex feasible region always occurs at one of the corner (vertex) points. Evaluate z = 4x + 6y at each corner point and identify which gives the minimum.
<p><strong>Step 1:</strong> Identify all corner points: (0, 2), (3, 0), (6, 0), (6, 8), (0, 5)</p><p><strong>Step 2:</strong> Evaluate z = 4x + 6y at each corner point:</p><ul><li>At (0, 2): z = 4(0) + 6(2) = 12</li><li>At (3, 0): z = 4(3) + 6(0) = 12</li><li>At (6, 0): z = 4(6) + 6(0) = 24</li><li>At (6, 8): z = 4(6) + 6(8) = 24 + 48 = 72</li><li>At (0, 5): z = 4(0) + 6(5) = 30</li></ul><p><strong>Step 3:</strong> The minimum value of z is 12, which occurs at both (0, 2) and (3, 0). The question asks where minimum occurs; depending on the options, the answer is typically one of these two points.</p><p>∴ Answer: D</p>
Correct Answer: D