Consider the function f(x) = 12x° + 45x* – 360x° + 2. f(x) has inflection points at (reading from left to right) x = %3D D, E, and F where D is and E is and F is For each of the following intervals, tell whether f(x) is concave up or concave down. (- 0, D): Select an answer (D, E): | Select an answer (E, F): | Select an answer (F, 0): Select an answer
Consider the function f(x) = 12x° + 45x* – 360x° + 2. f(x) has inflection points at (reading from left to right) x = %3D D, E, and F where D is and E is and F is For each of the following intervals, tell whether f(x) is concave up or concave down. (- 0, D): Select an answer (D, E): | Select an answer (E, F): | Select an answer (F, 0): Select an answer
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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![### Analyzing the Function \( f(x) \)
#### Given Function
Consider the function \( f(x) = 12x^5 + 45x^4 - 360x^3 + 2 \).
#### Inflection Points
\( f(x) \) has inflection points at (reading from left to right) \( x = D, E, \) and \( F \).
- **where \( D \) is:** [Input Field]
- **and \( E \) is:** [Input Field]
- **and \( F \) is:** [Input Field]
#### Concavity Assessment
For each of the following intervals, determine whether \( f(x) \) is concave up or concave down:
1. \( (-\infty, D) \) : [Dropdown] Select an answer
2. \( (D, E) \) : [Dropdown] Select an answer
3. \( (E, F) \) : [Dropdown] Select an answer
4. \( (F, \infty) \) : [Dropdown] Select an answer
### Explanation
- The given function \( f(x) = 12x^5 + 45x^4 - 360x^3 + 2 \) is a polynomial of degree five.
- Inflection points are where the concavity of the function changes; to find them, determine where the second derivative \( f''(x) \) changes sign.
- The concavity, either concave up or concave down, over specific intervals, is influenced by the behavior of \( f''(x) \).
This exercise involves finding the precise inflection points and analyzing the intervals for concavity, essential for understanding the behavior of the function graphically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf7777fa-aa3c-4ddb-8a8c-67588cdbbf49%2Fd6c80e8b-d604-42ed-b650-6d8c6f3a76a8%2Fa7qydal_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Analyzing the Function \( f(x) \)
#### Given Function
Consider the function \( f(x) = 12x^5 + 45x^4 - 360x^3 + 2 \).
#### Inflection Points
\( f(x) \) has inflection points at (reading from left to right) \( x = D, E, \) and \( F \).
- **where \( D \) is:** [Input Field]
- **and \( E \) is:** [Input Field]
- **and \( F \) is:** [Input Field]
#### Concavity Assessment
For each of the following intervals, determine whether \( f(x) \) is concave up or concave down:
1. \( (-\infty, D) \) : [Dropdown] Select an answer
2. \( (D, E) \) : [Dropdown] Select an answer
3. \( (E, F) \) : [Dropdown] Select an answer
4. \( (F, \infty) \) : [Dropdown] Select an answer
### Explanation
- The given function \( f(x) = 12x^5 + 45x^4 - 360x^3 + 2 \) is a polynomial of degree five.
- Inflection points are where the concavity of the function changes; to find them, determine where the second derivative \( f''(x) \) changes sign.
- The concavity, either concave up or concave down, over specific intervals, is influenced by the behavior of \( f''(x) \).
This exercise involves finding the precise inflection points and analyzing the intervals for concavity, essential for understanding the behavior of the function graphically.
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