Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.) f(x) 3x³ 3x² - 12x + 4 = (x, y) = Discuss the concavity of the graph of the function. (Enter your answers using interval notation.) concave upward concave downward Need Help? Read It
Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.) f(x) 3x³ 3x² - 12x + 4 = (x, y) = Discuss the concavity of the graph of the function. (Enter your answers using interval notation.) concave upward concave downward Need Help? Read It
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...
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![**Title: Finding the Point of Inflection and Discussing Concavity**
### Problem Statement:
Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.)
\[ f(x) = 3x^3 - 3x^2 - 12x + 4 \]
\[
(x, y) = \left( \text{\_\_\_\_\_\_, \_\_\_\_\_\_} \right)
\]
### Questions:
1. **Discuss the concavity of the graph of the function. (Enter your answers using interval notation.)**
- **Concave Upward:** \_\_\_\_\_\_
- **Concave Downward:** \_\_\_\_\_\_
2. **Need Help?**:
- **Read It** button (A clickable button for additional help).
**Explanation:**
- The function provided is a cubic polynomial \( f(x) = 3x^3 - 3x^2 - 12x + 4 \).
- You need to find the point(s) where the concavity of the function changes; these points are called points of inflection.
- The concavity is discussed in terms of intervals where the function is either concave upward or concave downward.
### Instructions:
1. **Finding the Point of Inflection:**
- To find the inflection points, determine where the second derivative of \( f(x) \) changes its sign.
- Compute the first and second derivatives of \( f(x) \).
- Solve \( f''(x) = 0 \) to find potential inflection points.
- Verify if these points are indeed points of inflection by checking the sign change in \( f''(x) \).
2. **Discuss Concavity:**
- Determine the intervals where the function is concave upward or concave downward based on the sign of the second derivative \( f''(x) \).
**Click "Read It" for more detailed explanations and step-by-step solutions.**
---
**Note:** This content is designed for educational purposes to assist in understanding calculus concepts related to polynomial functions, inflection points, and concavity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8a57943-65a4-4179-bc90-4de17f531e4f%2F2299db49-7d2a-4eba-b559-ad355dc100f2%2Fs5zbnbc_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Point of Inflection and Discussing Concavity**
### Problem Statement:
Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.)
\[ f(x) = 3x^3 - 3x^2 - 12x + 4 \]
\[
(x, y) = \left( \text{\_\_\_\_\_\_, \_\_\_\_\_\_} \right)
\]
### Questions:
1. **Discuss the concavity of the graph of the function. (Enter your answers using interval notation.)**
- **Concave Upward:** \_\_\_\_\_\_
- **Concave Downward:** \_\_\_\_\_\_
2. **Need Help?**:
- **Read It** button (A clickable button for additional help).
**Explanation:**
- The function provided is a cubic polynomial \( f(x) = 3x^3 - 3x^2 - 12x + 4 \).
- You need to find the point(s) where the concavity of the function changes; these points are called points of inflection.
- The concavity is discussed in terms of intervals where the function is either concave upward or concave downward.
### Instructions:
1. **Finding the Point of Inflection:**
- To find the inflection points, determine where the second derivative of \( f(x) \) changes its sign.
- Compute the first and second derivatives of \( f(x) \).
- Solve \( f''(x) = 0 \) to find potential inflection points.
- Verify if these points are indeed points of inflection by checking the sign change in \( f''(x) \).
2. **Discuss Concavity:**
- Determine the intervals where the function is concave upward or concave downward based on the sign of the second derivative \( f''(x) \).
**Click "Read It" for more detailed explanations and step-by-step solutions.**
---
**Note:** This content is designed for educational purposes to assist in understanding calculus concepts related to polynomial functions, inflection points, and concavity.
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