Explanation Given: The function is R ( x ) = − 0.6 x 3 + 3.7 x 2 + 5 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 ≤ 6 . Calculation: Evaluate R ′ ( x ) as follows. R ′ ( x ) = d d x ( − 0.6 x 3 + 3.7 x 2 + 5 x ) = − 0.6 ( 3 x 2 ) + 3.7 ( 2 x ) + 5 ( 1 ) = − 1.8 x 2 + 7.4 x + 5 Thus, R ′ ( x ) = − 1.8 x 2 + 7.4 x + 5 . The second derivative R ″ ( x ) can be obtained from R ′ ( x ) = − 1.8 x 2 + 7.4 x + 5 as follows. R ′ ′ ( x ) = d d x ( − 1.8 x 2 + 7.4 x + 5 ) = − 1.8 ( 2 x ) + 7.4 ( 1 ) = − 3.6 x + 7.4 Thus, the value of R ″ ( x ) is − 3.6 x + 7.4 . The point of diminishing returns occurs at the inflection point or R ″ ( x ) = 0 . So, equate R ″ ( x ) to 0. − 3.6 x + 7.4 = 0 3.6 x = 7.4 x = 7.4 3.6 x = 2.0556 x ≈ 2.06 To test the concavity at x = 2.06 divide the domain into two intervals ( 0 , 2.06 ) and ( 2.06 , 6 )

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Chapter 13.3, Problem 75E

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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 74E

Chapter 13.3, Problem 76E

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EBK FINITE MATHEMATICS AND CALCULUS WIT

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 (2.06,20.7) for the given function.

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