(a) Explanation Definition used: First derivative test: Suppose c to be a critical number for a function f . Assume that f is continuous on ( a , b ) and differentiable on ( a , b ) except at c , and c is the only critical number of f in ( a , b ) . f ( c ) is a relative maximum of f at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . f ( c ) is a relative minimum f at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Relative maximum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . Relative minimum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Calculation: The given cost function is C ( q ) = 100 + 20 q e − 0.01 q and the demand function is p = D ( q ) = 40 e − 0.01 q . Note that the revenue function R ( q ) is the product of the price per unit and the number of units sold. That is, R ( q ) = p × q Then the profit function is P ( q ) = R ( q ) − C ( q ) . Obtain the profit function as follows. P ( q ) = q p − C ( q ) = q ( 40 e − 0.01 q ) − ( 100 + 20 q e − 0.01 q ) = 40 q e − 0.01 q − 100 − 20 q e − 0.01 q = 20 q e − 0.01 q − 100 Therefore, the profit function is P ( q ) = 20 q e − 0.01 q − 100 . Obtain the derivative of the function as follows. P ′ ( q ) = ( 20 q e − 0 (b) To determine To find: The price of the unit p at which the maximum profit is attained. (c) To determine To find: The maximum profit for the given number of units.

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 13.2, Problem 45E

(a)

To determine

To find: The number of units, q that gives maximum profit when C(q)=100+20qe−0.01q and p=D(q)=40e−0.01q .

(b)

To determine

To find: The price of the unit p at which the maximum profit is attained.

(c)

To determine

To find: The maximum profit for the given number of units.

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Chapter 13.2, Problem 44E

Chapter 13.2, Problem 46E

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Finite Mathematics and Calculus with Applications

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The number of units, q that gives maximum profit when C ( q ) = 100 + 20 q e − 0.01 q and p = D ( q ) = 40 e − 0.01 q .

Solution Summary: The author explains that q is the number of units that gives maximum profit when C(q)=100+20q

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To find: The number of units, q that gives maximum profit when C(q)=100+20qe−0.01q and p=D(q)=40e−0.01q .

To find: The price of the unit p at which the maximum profit is attained.

To find: The maximum profit for the given number of units.

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Chapter 13 Solutions