Maximum Profit Function?

📌 CONCEPT: The maximum profit function is obtained by finding the critical points of the profit function, which is the difference between the revenue function and the cost function, and then determining the rate of change of the profit function at that point.

📐 RULE / FORMULA: To find the maximum profit function, we need to differentiate the profit function with respect to x, set it equal to zero, and solve for x. The profit function P(x) is given by P(x) = R(x) - C(x), where R(x) and C(x) are the revenue and cost functions, respectively.

💡 WORKED EXAMPLE: Let's find the maximum profit function for the given problem. The profit function is P(x) = R(x) - C(x) = (100x - x^2) - (2x^2 + 10x + 5) = -3x^2 - 10x - 5. To find the critical points, we differentiate P(x) with respect to x and set it equal to zero: dP/dx = -6x - 10 = 0. Solving for x, we get x = -10/6. We need to verify if this point corresponds to a maximum profit.

⚠️ COMMON MISTAKE: Students often forget to check the second derivative to confirm that the critical point corresponds to a maximum profit. They may also incorrectly differentiate the profit function or set the derivative equal to zero without considering the domain of the profit function.