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Optimal production

With a different price and a different cost function from the S1 example, the rule is the same. The profit is largest where the price equals the marginal cost. You solve that equation, insert the number into revenue minus cost, and get the profit in money. Because the cost has a positive coefficient of x squared, the marginal cost is increasing, and the point is a maximum of the profit.

Symbols

Example

Price 50 NOK and $C(x) = 0.1x^2 + 10x + 1000$.

$x = \dfrac{50 - 10}{0.2} = 200$. $P(200) = 10\,000 - 7000 = 3000$ NOK.

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Part of Mathematics S2: Cost, revenue and profit.