Given the nonlinear programming model: Max Z = 5x₁ - 2x₂² Subject to: x₁ + x₂ = 6 What is the optimal value of the objective function? A) Z = 19.625 B) Z = 20.625 C) Z = 21.625 D) Z = 22.625 E) Z =23.625

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Quizplus · Accepted Answer

To find the optimal value of the objective function, we first use the constraint x1 + x2 = 6 to express x1 in terms of x2 (or vice versa). Let's express x1 as x1 = 6 - x2. Substituting this into the objective function gives Z = 5(6 - x2) - 2x2^2 = 30 - 5x2 - 2x2^2. To maximize Z, we take the derivative of Z with respect to x2, set it to zero, and solve for x2. dZ/dx2 = -5 - 4x2 = 0 implies x2 = -5/-4 = 1.25. Substituting x2 = 1.25 into the constraint gives x1 = 6 - 1.25 = 4.75. Substituting x1 and x2 back into the objective function gives Z = 5(4.75) - 2(1.25)^2 = 23.75 - 3.125 = 20.625.

Given the Nonlinear Programming Model