Explanation Given information: The cost and revenue obtained after x items produced and sold is, C ( x ) = 4 x + 10 and R ( x ) = 50 x − 0.5 x 2 respectively. Where R ( x ) is revenue and C ( x ) is cost in dollars. Formula used: The concept of Max-Min Principle-2 is taken into consideration which is as follows: Suppose f is a function such that f ' ( x ) exists for every x in an interval l and there is exactly one critical value c in l , for which f ' ( c ) = 0 Then, f ( c ) is absolute maximum value over l if f ' ' ( c ) < 0 Or f ( c ) is absolute minimum value over l if f ' ' ( c ) > 0 Proof: The provided functions of revenue and cost are as follows: R ( x ) = 50 x − 0.5 x 2 C ( x ) = 4 x + 10 Subtract C ( x ) from R ( x ) , to get the profit P ( x ) : P ( x ) = R ( x ) − C ( x ) = 50 x − 0.5 x 2 − ( 4 x + 10 ) = − 0.5 x 2 + 46 x − 10 Now, first derivative of P ( x ) is, P ' ( x ) = − x + 46 The derivative exists for all values of x in [ 0 , ∞ )

Calculus and Its Applications (11th Edition)

11th Edition

ISBN: 9780321979391

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Chapter 2.5, Problem 78E

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To prove: The number of x units that must be produced at the cost of C(x)=4x+10 and sold at the price of R(x)=50x−0.5x2 in order to yield the maximum profit. Also calculate the maximum profit.

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Chapter 2.5, Problem 77E

Chapter 2.5, Problem 79E

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To prove: The number of x units that must be produced at the cost of C ( x ) = 4 x + 10 and sold at the price of R ( x ) = 50 x − 0.5 x 2 in order to yield the maximum profit. Also calculate the maximum profit.

Solution Summary: The author explains how the Max-Min Principle-2 functions of revenue and cost are taken into consideration.