Sets, Relations & Functions
Linear Programming
Grade 11
Question:
<p>In a linear programming problem, the objective function is \(z = 4x + 6y\). The corner points are (0, 2), (3, 0), (6, 0), (6, 8), and (0, 5). The minimum value of \(z\) occurs at:</p>
<p>Only at (0, 2)</p>
<p>Only at (3, 0)</p>
<p>At any point on the line segment joining (0, 2) and (3, 0)</p>
<p>At any point on the line segment joining (0, 2) and (0, 5)</p>
Step-by-Step Solution
Key Concept: In linear programming, the optimal value of a linear objective function always occurs at a corner point (vertex) of the feasible region. Evaluate the objective function at each corner point and compare to find the minimum.
<p><strong>Step 1:</strong> Evaluate z = 4x + 6y at each corner point:</p><p><strong>Step 2:</strong> At (0, 2): z = 4(0) + 6(2) = 0 + 12 = <strong>12</strong></p><p><strong>Step 3:</strong> At (3, 0): z = 4(3) + 6(0) = 12 + 0 = <strong>12</strong></p><p><strong>Step 4:</strong> At (6, 0): z = 4(6) + 6(0) = 24 + 0 = <strong>24</strong></p><p><strong>Step 5:</strong> At (6, 8): z = 4(6) + 6(8) = 24 + 48 = <strong>72</strong></p><p><strong>Step 6:</strong> At (0, 5): z = 4(0) + 6(5) = 0 + 30 = <strong>30</strong></p><p><strong>Step 7:</strong> Comparing all values: 12, 12, 24, 72, 30</p><p>The minimum value is z = 12, which occurs at both (0, 2) and (3, 0) (the entire edge between them gives minimum if answer requests single point, typically (0, 2) or (3, 0) depending on convention).</p><p>∴ Answer: C</p>
Correct Answer: C