Explain constrained optimization in management and solve a constrained minimization problem using the Lagrangian method.

Question One (Compulsory)

a) Constrained optimization in management involves finding the best possible outcome (e.g., maximum profit, minimum cost) for a business decision while adhering to specific limitations or restrictions. These limitations, such as budget, resources, or production capacity, are formally expressed as constraints that must be satisfied.

b) Given the constrained minimization problem:
Minimize $Z = 4x^2 + 2xy + y^2$
Subject to $x + y = 8$

i) Construct the corresponding Lagrangian function.
The objective function is $f(x, y) = 4x^2 + 2xy + y^2$.
The constraint is $g(x,