Answer :
The minimum point of the objective function is (x₁, x₂, x₃, x₄) = (-5, 3, 2, -4).
To find the minimum point, we can use the method of Lagrange multipliers. Let's define the Lagrangian function L as:
L(x₁, x₂, x₃, x₄, λ₁, λ₂) = x₁x₃ + x₂x₄ + 11x₃ + 28x₄ + 8 - λ₁(x₁ + 3x₂ - 19x₃ - 16x₄ - 27) - λ₂(-2x₁ - 5x₂ + 32x₃ + 26x₄ + 46)
We want to minimize L with respect to x₁, x₂, x₃, and x₄, and satisfy the given constraints. Taking the partial derivatives of L with respect to x₁, x₂, x₃, and x₄, and setting them equal to zero, we get the following system of equations:
∂L/∂x₁ = x₃ - λ₁ - 2λ₂ = 0 ...(1)
∂L/∂x₂ = x₄ + 3λ₁ - 5λ₂ = 0 ...(2)
∂L/∂x₃ = x₁ + 11 - 19λ₁ + 32λ₂ = 0 ...(3)
∂L/∂x₄ = x₂ + 28 - 16λ₁ + 26λ₂ = 0 ...(4)
We also need to satisfy the constraint equations:
x₁ + 3x₂ - 19x₃ - 16x₄ = 27 ...(5)
-2x₁ - 5x₂ + 32x₃ + 26x₄ = -46 ...(6)
Solving this system of equations, we find that x₁ = -5, x₂ = 3, x₃ = 2, x₄ = -4.
Therefore, the minimum point of the objective function is (x₁, x₂, x₃, x₄) = (-5, 3, 2, -4).
To know more about Lagrange multipliers, refer here:
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