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A collection of optimization problems in calculus, where the goal is to find maximum or minimum values of functions subject to certain restrictions. The problems cover various scenarios, such as finding two numbers with given sum and product, maximizing volume of a box, and determining the price for maximum profit. The extreme value theorem, first derivative test, and second derivative test are used to solve these problems.
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Many application problems in calculus involve functions for which you want to find maximum or minimum values. The restrictions stated or implied for such functions will determine the domain from which you must work. The function, together with its domain, will suggest which technique is appropriate to use in determining a maximum or minimum value - the Extreme Value Theorem, the First Derivative Test, or the Second Derivative Test.
2 and all units can be sold at a price of p(x) = 49 - x dollars per unit. Determine the price that corresponds to the maximum profit.