Explanation Given: The function is R ( x ) = 10 , 000 − x 3 + 42 x 2 + 800 x , where R ( x ) is the revenue calculated in thousand dollars and x is the amount spent on advertising. The range of x is 0 ≤ x ≤ 20 . Calculation: Evaluate R ′ ( x ) as follows. R ′ ( x ) = d d x ( 10 , 000 − x 3 + 42 x 2 + 800 x ) = − 3 x 2 + 42 ( 2 x ) + 800 = − 3 x 2 + 84 x + 800 Thus, R ′ ( x ) = − 3 x 2 + 84 x + 800 . The second derivative R ″ ( x ) can be obtained from R ′ ( x ) = − 3 x 2 + 84 x + 800 as follows. R ′ ′ ( x ) = d d x ( − 3 x 2 + 84 x + 800 ) = − 3 ( 2 x ) + 84 ( 1 ) = − 6 x + 84 Thus, the value of R ″ ( x ) is − 6 x + 84 . The point of diminishing returns occurs at the inflection point or R ″ ( x ) = 0 . So, equate R ″ ( x ) to 0. − 6 x + 84 = 0 6 x = 84 x = 84 6 x = 14 To test the concavity at x = 14 divide the domain into two intervals ( 0 , 14 ) and ( 14 , 20 ) . Take a test point in each of the intervals and find the value of R ″ ( x )

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 13.3, Problem 72E

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To find: The point of diminishing returns (x,y) for the given function.

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Chapter 13.3, Problem 71E

Chapter 13.3, Problem 73E

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Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

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The point of diminishing returns ( x , y ) for the given function.

Solution Summary: The author explains that the point of diminishing returns is (x,y) for the given function.

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