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

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 13.3, Problem 72E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.3, Problem 71E

Chapter 13.3, Problem 73E

Students have asked these similar questions

Applying the Second Derivative Test Find the critical points of f (x, y) = (x2 + y2)e-x and analyze them using the Second Derivative Test.

3.USING GRAPHPAD PRISM: The transformation rate you observed in a treated group of 1036 yeast colonies is 0.193%, while the rate among 1441 untreated colonies is 0.555 %, is there an effect of treatment on transformation? Determine the p-value. SHOW YOUR WORk

Analyzing critical points Use the Second Derivative Test to classify the critical points of ƒ(x, y) = x2 + 2y2 - 4x + 4y + 6.

Chapter 13 Solutions

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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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.

Concept explainers

Videos

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

Want to see the full answer?

Chapter 13 Solutions