The feasible region for a set of constraints has vertices at \( (2,0),(10,1),(8,5) \), and ( 0 , 4). Given this feasible region, find the maximum value of the objective function. \( E=2 x-3 y \) 1 -12 17 4

To find the maximum value of the objective function \( E = 2x - 3y \) at the vertices of the feasible region, we evaluate \( E \) at each vertex: - At \( (2,0) \): \( E = 2(2) - 3(0) = 4 \) - At \( (10,1) \): \( E = 2(10) - 3(1) = 20 - 3 = 17 \) - At \( (8,5) \): \( E = 2(8) - 3(5) = 16 - 15 = 1 \) - At \( (0,4) \): \( E = 2(0) - 3(4) = -12 \) The maximum value among these is \( 17 \) at the vertex \( (10,1) \). In linear programming, finding maximum values at vertices is important because the solution to an optimization problem will occur at one of the feasible region's vertices. By systematically testing vertex points, you can efficiently locate the global optimum without exhaustive exploration of every possible point in the feasible region. Keep an eye on feasible boundaries!