Explanation Formula used : Power rule of differentiation d d x ( x n ) = n x n − 1 The general form of a quadratic equation is a x 2 + b x + c = 0 The solution of above equation is given by x = − b ± b 2 − 4 a c 2 a . Given: The equation that relates price function to quantity demanded is p ( x ) = − 0.05 x + 600 . Where, x is the number of televisions produced. The revenue generated by sale of x televisions per week is given by, R ( x ) = x × p ( x ) The equation that relates weekly total cost (in dollars) for manufacturing with televisions produced is given by, C ( x ) = 0.000002 x 3 − 0.03 x 2 + 400 x + 80 , 000 Calculation : Revenue function is defines as product of number of units produced to the price of each unit R ( x ) = x × p ( x ) . It is given that, the equation relates price function to quantity demanded is p ( x ) = − 0.05 x + 600 . Substitute p ( x ) = − 0.05 x + 600 in R ( x ) = x × p ( x ) , R ( x ) = x ( − 0.05 x + 600 ) = − 0.05 x 2 + 600 x Profit function is defined as the difference of revenue function and total cost function. Take P ( x ) as profit function such that P ( x ) = R ( x ) − C ( x ) Substitute − 0.05 x 2 + 600 x for R ( x ) and 0.000002 x 3 − 0.03 x 2 + 400 x + 80 , 000 for C ( x ) in above equation to obtain the production function P ( x ) . P ( x ) = − 0.05 x 2 + 600 x − ( 0.000002 x 3 − 0.03 x 2 + 400 x + 80000 ) = − 0.05 x 2 + 600 x − 0.000002 x 3 + 0.03 x 2 − 400 x − 80000 That is, P ( x ) = − 0.000002 x 3 − 0.02 x 2 + 200 x − 80 , 000 (1) First derivative test: Equate the first derivative of an equation to zero; it gives the point of extrema (either maxima or minima). Use power rule to differentiate P ( x ) given in equation (1) with respect to x . P ′ ( x ) = d d x ( − 0

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Chapter 4.4, Problem 50E

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The number of televisions manufacturer should produce to maximize the profit.

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Chapter 4.4, Problem 49E

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Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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The number of televisions manufacturer should produce to maximize the profit.

Solution Summary: The author explains that the manufacturer should produce 3,333 televisions to maximize the profit. The equation relates price function to quantity demanded.

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