Maximising Revenue?

📌 CONCEPT: To maximize revenue, we need to find the optimal production level of a product that balances the selling price and the cost price per unit, given by the revenue function R(x).

📐 RULE / FORMULA: The revenue function is given by R(x) = (selling price per unit) × (number of units sold) = (100 - 2x)x.

💡 WORKED EXAMPLE: Let's find the maximum revenue function for the given problem. The revenue function is R(x) = (100 - 2x)x = -2x^2 + 100x. To find the maximum revenue, we need to find the critical points of the function. We can do this by finding the derivative of R(x) with respect to x, setting it equal to zero, and solving for x. The derivative of R(x) is R'(x) = -4x + 100. Setting R'(x) = 0, we get -4x + 100 = 0, which gives x = 25. This is the critical point where the revenue is maximized.

⚠️ COMMON MISTAKE: Students often forget to check the second derivative to ensure that the critical point corresponds to a maximum. They should also ensure that the revenue function is a quadratic function with a negative leading coefficient to guarantee a maximum value.