Sets, Relations & Functions
Linear Programming
Grade None
Question:
<p>In a linear programming problem, the objective function is \(z = 4x + 3y\). The corner points of the feasible region are (0, 8), (2, 5), (4, 3), and (9, 0). Find the minimum value of \(z\).</p>
<p>24</p>
<p>23</p>
<p>25</p>
<p>36</p>
Step-by-Step Solution
Key Concept: In linear programming, the optimal value of a linear objective function over a convex feasible region always occurs at a corner point. Evaluate the objective function at all corner points and compare.
<p><strong>Step 1:</strong> Identify that for a linear objective function z = 4x + 3y over a convex feasible region, the extremum occurs at a corner point.</p><p><strong>Step 2:</strong> Evaluate z at each corner point:</p><ul><li>At (0, 8): z = 4(0) + 3(8) = 0 + 24 = 24</li><li>At (2, 5): z = 4(2) + 3(5) = 8 + 15 = 23</li><li>At (4, 3): z = 4(4) + 3(3) = 16 + 9 = 25</li><li>At (9, 0): z = 4(9) + 3(0) = 36 + 0 = 36</li></ul><p><strong>Step 3:</strong> Compare all values: {24, 23, 25, 36}. The minimum value is 23, occurring at the corner point (2, 5).</p><p>∴ <strong>Answer: B (Minimum value = 23)</strong></p>
Correct Answer: B