Maximizing Revenue: An E-commerce Dilemma?

📌 CONCEPT: The problem involves finding the optimal price at which the revenue is maximized using the demand function, given the initial price and the rate of decrease in demand due to market competition.

📐 RULE / FORMULA: To solve this problem, we use the concept of optimization of revenue, which is given by the product of the demand function and the price of the product, and we maximize it using the first derivative test.

💡 WORKED EXAMPLE: Let's consider the demand function P(x) = 100(1 - 0.02)^x, where P(x) is the price at day x. To maximize revenue, we need to find the critical points of the revenue function R(x) = P(x) * x. Taking the derivative of R(x) with respect to x gives us R'(x) = P(x) + xP'(x). Setting R'(x) = 0, we can solve for the optimal price x that maximizes revenue.

⚠️ COMMON MISTAKE: Students often make the mistake of ignoring the second derivative test to confirm whether the critical point corresponds to a maximum or minimum, which may lead to incorrect conclusions in optimization problems.