Let f(x) = 2x²(x - 7) + 12. Find the critical points c that correspond to local minima. (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.) C = Find the critical points c that correspond to local maxima. (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.) C = Find values at which the points of inflection occur. (Use symbolic notation and fractions where needed. Give your answer as a comma separated list. Enter DNE if there are no points of inflection.)
Let f(x) = 2x²(x - 7) + 12. Find the critical points c that correspond to local minima. (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.) C = Find the critical points c that correspond to local maxima. (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.) C = Find values at which the points of inflection occur. (Use symbolic notation and fractions where needed. Give your answer as a comma separated list. Enter DNE if there are no points of inflection.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Set: Critical Points and Points of Inflection
#### Given Function:
Let \( f(x) = 2x^2(x - 7) + 12 \).
---
### Question 1: Local Minima
**Find the critical points \( c \) that correspond to local minima.**
*Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.*
\[ c = \text{_____________________} \]
---
### Question 2: Local Maxima
**Find the critical points \( c \) that correspond to local maxima.**
*Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.*
\[ c = \text{_____________________} \]
---
### Question 3: Points of Inflection
**Find values at which the points of inflection occur.**
*Use symbolic notation and fractions where needed. Give your answer as a comma separated list. Enter DNE if there are no points of inflection.*
\[ c = \text{_____________________} \]
---
**Instructions for Submission:**
Please ensure that all answers are given in symbolic notation when required and that all lists of answers are comma-separated. If there are no critical points or points of inflection for a particular part, indicate this by writing 'DNE' (Does Not Exist).
**Note:** This problem set is intended to help you understand how to find the properties of a given function, including its critical points and points of inflection, by using calculus methods such as finding the derivatives and solving for zeros.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0c43ed0-a854-4f55-a30b-147647d0ed33%2F0c43639b-aca7-45d5-8b04-c1956d221c1c%2Fblgjbtd_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Set: Critical Points and Points of Inflection
#### Given Function:
Let \( f(x) = 2x^2(x - 7) + 12 \).
---
### Question 1: Local Minima
**Find the critical points \( c \) that correspond to local minima.**
*Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.*
\[ c = \text{_____________________} \]
---
### Question 2: Local Maxima
**Find the critical points \( c \) that correspond to local maxima.**
*Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.*
\[ c = \text{_____________________} \]
---
### Question 3: Points of Inflection
**Find values at which the points of inflection occur.**
*Use symbolic notation and fractions where needed. Give your answer as a comma separated list. Enter DNE if there are no points of inflection.*
\[ c = \text{_____________________} \]
---
**Instructions for Submission:**
Please ensure that all answers are given in symbolic notation when required and that all lists of answers are comma-separated. If there are no critical points or points of inflection for a particular part, indicate this by writing 'DNE' (Does Not Exist).
**Note:** This problem set is intended to help you understand how to find the properties of a given function, including its critical points and points of inflection, by using calculus methods such as finding the derivatives and solving for zeros.
![### Determine the Interval of Concavity
#### Determine the interval on which \( f \) is concave up.
(Use symbolic notation and fractions where needed. Give your answer as an interval in the form \( (\ast, \ast) \). Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and an appropriate type of parenthesis "\( (\), \( ) \)" or "\( [ \), \( ] \)" depending on whether the interval is open or closed. Enter \( \emptyset \) if the interval is empty.)
\( x \in \)
[Input box here]
#### Determine the interval on which \( f \) is concave down.
(Use symbolic notation and fractions where needed. Give your answer as an interval in the form \( (\ast, \ast) \). Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and an appropriate type of parenthesis "\( (\), \( ) \)" or "\( [ \), \( ] \)" depending on whether the interval is open or closed. Enter \( \emptyset \) if the interval is empty.)
\( x \in \)
[Input box here]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0c43ed0-a854-4f55-a30b-147647d0ed33%2F0c43639b-aca7-45d5-8b04-c1956d221c1c%2F33i3sch_processed.png&w=3840&q=75)
Transcribed Image Text:### Determine the Interval of Concavity
#### Determine the interval on which \( f \) is concave up.
(Use symbolic notation and fractions where needed. Give your answer as an interval in the form \( (\ast, \ast) \). Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and an appropriate type of parenthesis "\( (\), \( ) \)" or "\( [ \), \( ] \)" depending on whether the interval is open or closed. Enter \( \emptyset \) if the interval is empty.)
\( x \in \)
[Input box here]
#### Determine the interval on which \( f \) is concave down.
(Use symbolic notation and fractions where needed. Give your answer as an interval in the form \( (\ast, \ast) \). Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and an appropriate type of parenthesis "\( (\), \( ) \)" or "\( [ \), \( ] \)" depending on whether the interval is open or closed. Enter \( \emptyset \) if the interval is empty.)
\( x \in \)
[Input box here]
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