If the minimum value of an objective function Z = ax + by occurs at two points (3, 4) and (4, 3), then
An objective function in linear programming is a mathematical expression that represents the goal of the optimization problem. It defines what needs to be maximized or minimized, such as profit or cost. The objective function is subject to constraints and is optimized within the feasible region to find the best solution.
Linear Programming is a technique used to optimize a linear objective function subject to linear constraints. It helps in making optimal decisions by finding the best solution within a feasible region. The feasible region is formed by the constraints and the optimal solution is usually found at the vertices of this region.
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Given the objective function Z = ax + by, the minimum value occurs at two points (3, 4) and (4, 3).
1. At point (3, 4), the objective function is Z = a(3) + b(4) = 3a + 4b.
2. At point (4, 3), the objective function is Z = a(4) + b(3) = 4a + 3b.
Since both points give the minimum value of Z, the objective function at these points must be equal:
3a + 4b = 4a + 3b
Solving for a and b:
3a + 4b – 4a – 3b = 0
-a + b = 0 ⟹ a = b
Thus, the correct condition is a = b.
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