Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter3: Linear And Nonlinear Functions
Section3.3: Rate Of Change And Slope
Problem 19PPS
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.
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