Find the point of diminishing returns (x,y) for the function R(x), where R(x) represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars). R(x) = 11,000-x³ +45x² +800x, 0≤x≤20 The point of diminishing returns is (Type an ordered pair.)
Find the point of diminishing returns (x,y) for the function R(x), where R(x) represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars). R(x) = 11,000-x³ +45x² +800x, 0≤x≤20 The point of diminishing returns is (Type an ordered pair.)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
ChapterP: Prerequisites
SectionP.7: A Library Of Parent Functions
Problem 47E
Related questions
Question
100%
![### Find the Point of Diminishing Returns for the Function
**Problem Statement:**
Find the point of diminishing returns (x,y) for the function \( R(x) \), where \( R(x) \) represents revenue (in thousands of dollars), and \( x \) represents the amount spent on advertising (in thousands of dollars).
Function:
\[ R(x) = 11,000 - x^3 + 45x^2 + 800x, \quad 0 \leq x \leq 20 \]
**Question:**
The point of diminishing returns is:
\[ \_\_\_\_\_\_\_ \]
(Type an ordered pair.)
**Explanation:**
1. **Definition**: The point of diminishing returns refers to the point at which the rate of revenue growth starts to decline even though more is being spent on advertising.
2. **Approach**: To determine this point, we need to compute the derivative of \( R(x) \) and find the point where the second derivative changes sign, indicating a maximum or a point of inflection.
**Steps to Solve:**
1. Compute the first derivative \( R'(x) \).
2. Compute the second derivative \( R''(x) \).
3. Find the critical points by setting \( R'(x) \) to zero.
4. Determine the points of diminishing returns by analyzing the second derivative.
**Note**: This is an exercise in finding and understanding the mathematical concept of diminishing returns using calculus and critical thinking. The exact numerical solution requires further computation, which is part of the learning process in applying these mathematical principles.
**Interactive input:**
Type an ordered pair to submit your answer after solving for x and y values where the point of diminishing returns occurs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46a4dff8-7f9e-4127-9d1e-ef0349b76ae2%2Fd33eac9f-9c9e-4d3c-80b7-271958947e1e%2F1yugsda_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Find the Point of Diminishing Returns for the Function
**Problem Statement:**
Find the point of diminishing returns (x,y) for the function \( R(x) \), where \( R(x) \) represents revenue (in thousands of dollars), and \( x \) represents the amount spent on advertising (in thousands of dollars).
Function:
\[ R(x) = 11,000 - x^3 + 45x^2 + 800x, \quad 0 \leq x \leq 20 \]
**Question:**
The point of diminishing returns is:
\[ \_\_\_\_\_\_\_ \]
(Type an ordered pair.)
**Explanation:**
1. **Definition**: The point of diminishing returns refers to the point at which the rate of revenue growth starts to decline even though more is being spent on advertising.
2. **Approach**: To determine this point, we need to compute the derivative of \( R(x) \) and find the point where the second derivative changes sign, indicating a maximum or a point of inflection.
**Steps to Solve:**
1. Compute the first derivative \( R'(x) \).
2. Compute the second derivative \( R''(x) \).
3. Find the critical points by setting \( R'(x) \) to zero.
4. Determine the points of diminishing returns by analyzing the second derivative.
**Note**: This is an exercise in finding and understanding the mathematical concept of diminishing returns using calculus and critical thinking. The exact numerical solution requires further computation, which is part of the learning process in applying these mathematical principles.
**Interactive input:**
Type an ordered pair to submit your answer after solving for x and y values where the point of diminishing returns occurs.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill