Optimization of a Function

📌 CONCEPT: To maximize the total profit, we need to find the values of x and y that maximize the function P(x, y) = f(x) + g(y), where f(x) and g(y) represent the profits from products A and B, respectively.

📐 RULE / FORMULA: We will use the method of partial derivatives to find the critical points of the function P(x, y). This involves finding the partial derivatives of P with respect to x and y, and setting them equal to zero.

💡 WORKED EXAMPLE: Let's find the critical points of the function P(x, y) = (2x^2 - 12x + 15) + (-x^2 + 6x + 5) = x^2 - 6x + 20. The partial derivatives are P_x = 2x - 6 and P_y = 0 (since there is no y term). Setting P_x = 0, we get 2x - 6 = 0, which gives x = 3. Substituting x = 3 into P(x, y), we get P(3, y) = 4 - 6 + 20 = 18. Since there is no y term, y can be any value.

⚠️ COMMON MISTAKE: Students often forget to check the second-order partial derivatives to confirm that the critical point is a maximum. In this case, we would need to check that P_xx < 0 and P_xy = P_yx = 0.