# A retired potter can produce china pitchers at a cost of $5.00 each. Shes estimates her price function to be p=17-0.5x, where p is the price at which exactly x pitchers will be sold per week....

A retired potter can produce china pitchers at a cost of $5.00 each. Shes estimates her price function to be p=17-0.5x, where p is the price at which exactly x pitchers will be sold per week. Find the number of pitchers that shw should produce and the price that she should charge to maximize profit, and find her maximum profit.

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The potter can produce pitchers at a cost of $5. The price function is p = 17 - 0.5*x where x is the number of pitchers sold in a week. The profit made by the potter when x pitchers are sold is P = p*x - 5*x = (17 - 0.5*x)*x - 5x = 17x - 0.5x^2 - 5x = 12x - 0.5*x^2

To maximize profit, the number of pitchers that should be sold is given by the solution of P' = 0

=> 12 - x = 0

=> x = 12

The price of the pitchers at which 12 are sold is 17 - 6 = 11. The maximum profit made by the potter is $72

**To maximize profits, the potter should charge $11 per pitcher and the maximum profit is $72**