Final answer:
The question asks to explain the marginal profit function for a record club, which involves analyzing the relationship between output and profit maximization based on marginal revenue and marginal cost for a given number of club members.
Step-by-step explanation:
The student's question pertains to the concept of marginal profit in economics, which is a part of the high school mathematics curriculum, typically covered under functions and calculus-related topics. The marginal profit, P'(x), described in the question is represented by a cubic function, and it models the additional profit earned by increasing the number of members in the record club by one. A crucial point in understanding marginal profit is that it allows a company to determine at what level of output (or membership, in this case) they maximize profit. For instance, if we have an output of 4 units where marginal revenue is 600 and marginal cost is 250, the company would clearly increase overall profits by producing this unit. However, at an output of 5 units where marginal revenue equals marginal cost, there is no change in profits. Practical decisions on the number of members to take would similarly be based on the formula provided in the question.
When analyzing such functions, it's also key to understand the relationship between total revenue, total cost, and profits. Total profits are calculated by subtracting total costs from total revenues. This involves not just the marginal values but the actual costs and revenues at a given level of output. For instance, producing a certain number of units might result in total revenues of $640 and total costs of $580, leading to profits of $60. Here, the firm is making economic profits because the price lies above the average cost curve.