Q.A manufacturer can sell $x$ items at a price of ₹$(280 - x)$ each. The cost of producing $x$ items is ₹$(x^2 + 40x + 35)$. Find the number of items to be sold so that the manufacturer can make maximum profit.

Profit $\pi(x) = R(x) - C(x)$ is maximised where $\pi'(x) = \text{MR} - \text{MC} = 0$, i.e. marginal revenue equals marginal cost, confirmed by $\pi''(x) < 0$. Average cost $\overline{C}(x) = \dfrac{C(x)}{x}$ is minimised where $\overline{C}'(x) = 0$ with $\overline{C}''(x) > 0$, and at that output average cost equals marginal cost. Revenue alone is maximised where $\text{MR} = 0$. Every optim …