Maximizing Profit under Constraints?

📌 CONCEPT: The problem of maximizing profit under constraints is solved using the method of Lagrange multipliers, which allows us to find the optimal values of the variables subject to given constraints.

📐 RULE / FORMULA: The Lagrangian function is defined as L(x, y, λ) = f(x, y) - λ(g(x, y) - d), where f(x, y) is the profit function, g(x, y) is the constraint function, λ is the Lagrange multiplier, and d is the maximum value of the constraint.

💡 WORKED EXAMPLE: Suppose a company manufactures x units of product A and y units of product B. The profit is Rs. 10 per unit of product A and Rs. 15 per unit of product B. It takes 2 hours to produce one unit of product A and 3 hours to produce one unit of product B. The machines are available for at most 120 hours. Let x be the number of units of product A and y be the number of units of product B. The profit function is P(x, y) = 10x + 15y. The constraint function is 2x + 3y ≤ 120. We want to maximize the profit subject to the constraint.

⚠️ COMMON MISTAKE: Students often forget to consider the equality case when solving the system of equations, which may lead to incorrect solutions.