(a) Explanation Given: Demand function is p ( x ) = 124 − 2 x . Total cost is C ( x ) = 2 x 3 − 59 x 2 + 4 x + 76 . Calculation: Consider the following function for the parts of a piece of a heavy machine, p ( x ) = 124 − 2 x ...... ( 1 ) And, the following function represents the total cost, C ( x ) = 2 x 3 − 59 x 2 + 4 x + 76 ...... ( 2 ) . The profit function will be, p ( x ) = x p ( x ) − C ( x ) = x [ 124 − 2 x ] − [ 2 x 3 − 59 x 2 + 4 x + 76 ] = 124 x − 2 x 2 − 2 x 3 + 59 x 2 − 4 x − 76 = − 2 x 3 + 57 x 2 + 120 x − 76 As the profit function is a polynomial function, it will be continuous for all x . Differentiate it with respect to x . p ' ( x ) = d ( − 2 x 3 + 57 x 2 + 120 x − 76 ) d x = − 2 × 3 x 2 + 57 × 2 x + 120 × 1 − 0 = − 6 x 2 + 114 x + 120 = − 6 ( x 2 − 19 x − 20 ) Substitute P ' ( x ) = 0 and solve for x (b) To determine To find: the consumer’s surplus with respect to the price that corresponds to maximum profit.

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Question

Chapter 6.6, Problem 29PS

(a)

To determine

To find: the maximum number of profits.

(b)

To determine

To find: the consumer’s surplus with respect to the price that corresponds to maximum profit.

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Chapter 6.6, Problem 28PS

Chapter 6.6, Problem 30PS

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Chapter 6 Solutions

Calculus

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the maximum number of profits.

Solution Summary: The author explains that the profit function is a polynomial function, it will be continuous for all x.