You want to sell a certain number {{MATH_51}} of items in order to maximize your profit. Market research tells you that if you set the price at {{CMD_52}}{{MATH_53}}, you will be able to sell {{MATH_54}} items, and for every {{MATH_55}} cents you lower the price below {{CMD_56}}{{MATH_57}} you will be able to sell another {{MATH_58}} items. Suppose that your fixed costs (``start-up costs'') total {{CMD_59}}{{MATH_60}}, and the per item cost of production (``marginal cost'') is {{CMD_61}}{{MATH_62}}. Find the price to set per item and the number of items sold in order to maximize profit, and also determine the maximum profit you can get. {{CMD_63}} The first step is to convert the problem into a function maximization problem. The revenue for selling {{MATH_64}} items at {{MATH_65}} dollars is given by {{MATH_66}} and the cost of producing {{MATH_67}} items is given by {{MATH_68}} However, from the problem we see that the number of items sold is itself a function of {{MATH_69}}, {{MATH_70}} So profit is give by: {{ENV_71}} We want to know the maximum value of the function {{MATH_72}} on the interval {{MATH_73}}.