Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Title: Finding the Points of Inflection for Various Functions**
**Introduction**
In calculus, the points of inflection of a function are the points where the function changes concavity, which means the curve shifts from being concave up to concave down or vice versa. To find these points, we typically analyze the second derivative of the function. Below, we have three different functions with the task of finding their points of inflection.
**Problems**
**14) Given Function:**
\[ f(x) = 7e^{-x^2} \]
*To find the points of inflection:*
1. Compute the first derivative \( f'(x) \).
2. Compute the second derivative \( f''(x) \).
3. Set \( f''(x) = 0 \) and solve for \( x \) to find potential points of inflection.
4. Verify the change in concavity by analyzing the sign of \( f''(x) \) around the potential points.
**15) Given Function:**
\[ f(x) = x^3 + 24x^2 + 189x + 488 \]
*To find the points of inflection:*
1. Compute the first derivative \( f'(x) \).
2. Compute the second derivative \( f''(x) \).
3. Set \( f''(x) = 0 \) and solve for \( x \) to find potential points of inflection.
4. Verify the change in concavity by analyzing the sign of \( f''(x) \) around the potential points.
**16) Given Function:**
\[ f(x) = x^2 + 2x + 2 \]
*To find the points of inflection:*
1. Compute the first derivative \( f'(x) \).
2. Compute the second derivative \( f''(x) \).
3. Set \( f''(x) = 0 \) and solve for \( x \). However, if the second derivative is a constant (non-zero), the function does not have any points of inflection.
**Conclusion**
By following these steps, one can determine the points at which each function changes concavity, thus identifying the points of inflection. Each step involves computing derivatives and setting the second derivative to zero to find critical points. Remember to verify these points by checking the sign change in the second derivative around the identified values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b6dc6c4-2828-4529-a1c7-3495f006d86b%2Ff9d556dc-0ac4-4e02-9b26-92c0d1ab5b9c%2Fzxm8gm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Points of Inflection for Various Functions**
**Introduction**
In calculus, the points of inflection of a function are the points where the function changes concavity, which means the curve shifts from being concave up to concave down or vice versa. To find these points, we typically analyze the second derivative of the function. Below, we have three different functions with the task of finding their points of inflection.
**Problems**
**14) Given Function:**
\[ f(x) = 7e^{-x^2} \]
*To find the points of inflection:*
1. Compute the first derivative \( f'(x) \).
2. Compute the second derivative \( f''(x) \).
3. Set \( f''(x) = 0 \) and solve for \( x \) to find potential points of inflection.
4. Verify the change in concavity by analyzing the sign of \( f''(x) \) around the potential points.
**15) Given Function:**
\[ f(x) = x^3 + 24x^2 + 189x + 488 \]
*To find the points of inflection:*
1. Compute the first derivative \( f'(x) \).
2. Compute the second derivative \( f''(x) \).
3. Set \( f''(x) = 0 \) and solve for \( x \) to find potential points of inflection.
4. Verify the change in concavity by analyzing the sign of \( f''(x) \) around the potential points.
**16) Given Function:**
\[ f(x) = x^2 + 2x + 2 \]
*To find the points of inflection:*
1. Compute the first derivative \( f'(x) \).
2. Compute the second derivative \( f''(x) \).
3. Set \( f''(x) = 0 \) and solve for \( x \). However, if the second derivative is a constant (non-zero), the function does not have any points of inflection.
**Conclusion**
By following these steps, one can determine the points at which each function changes concavity, thus identifying the points of inflection. Each step involves computing derivatives and setting the second derivative to zero to find critical points. Remember to verify these points by checking the sign change in the second derivative around the identified values.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning