A manufacturer sells lamps at six dollars each and sells 3000 each month. for each one dollar that the price is increased, 1000 fewer lamps are sold each month. it costs 1 dollars to make one lamp. what price should lamps be sold at to maximize profit?

Let x = price increase (in dollars)

Then the revenue is (6+x) * (3000 - 1000x) and the cost is (3000 - 1000x).

thus the profit P = revenue minus cost = 18,000 - 6000x^2 - 3000 + 1000x = 15,000 + 1,000 - 6,000x^2

take the derivative and set it equal to zero and I think the price increase x should be 1/12 so the final price would be original price plus price increase or 6 + 1/12