Use the graph to find the point(s) of inflection of f. (Give your answer as a comma-separated list of points in the form (*, *). Use decimal notation. Give your numbers to three decimal places. Enter DNE if the function has no inflection points.) inflection point(s): | Use the graph to determine the interval(s) on which the function ƒ is concave down. (Give your answer as an interval in the form (*, *). Use the symbol oo for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Use decimal notation. Give your numbers to three decimal places.)
Use the graph to find the point(s) of inflection of f. (Give your answer as a comma-separated list of points in the form (*, *). Use decimal notation. Give your numbers to three decimal places. Enter DNE if the function has no inflection points.) inflection point(s): | Use the graph to determine the interval(s) on which the function ƒ is concave down. (Give your answer as an interval in the form (*, *). Use the symbol oo for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Use decimal notation. Give your numbers to three decimal places.)
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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![**Instructions: Using the graphing utility, graph the function \( y = f'(x) \).**
### Function:
\[ f'(x) = x^4 - 16x^3 + 91x^2 - 216x + 180 \]
### Graph:
- **Axes:** The graph is plotted on a standard Cartesian plane with the horizontal axis labeled as \( x \) and the vertical axis labeled as \( y \).
- **Range:** The \( x \)-axis ranges from \(-40\) to \(20\), and the \( y \)-axis ranges from \(-400\) to \(400\).
- **Graph Behavior:**
- The graph of the derivative function \( f'(x) \) dips sharply around \( x = 0 \).
- The curve then rises steeply after this dip.
- There is a visible local minimum close to the origin, indicating a shift in the curve's direction.
### Graphing Controls:
- The graph is implemented using Desmos, a graphing tool providing zoom in/out features, represented by plus (+) and minus (−) buttons.
- A home button is also present to reset the view.
### Description:
The function \( f'(x) = x^4 - 16x^3 + 91x^2 - 216x + 180 \) is a polynomial of degree 4, and its graph shows typical polynomial behavior with varying slopes and curvature. The graph is characterized by its steep decline and subsequent rise near the y-axis, indicative of changes in the slope and curvature of the original function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bd5995a-c4b2-468d-a13c-4876a73e2cf6%2F17da13cd-6f70-490d-9de6-e7ab55e7b0f3%2Fpsjpiwo_processed.png&w=3840&q=75)
Transcribed Image Text:**Instructions: Using the graphing utility, graph the function \( y = f'(x) \).**
### Function:
\[ f'(x) = x^4 - 16x^3 + 91x^2 - 216x + 180 \]
### Graph:
- **Axes:** The graph is plotted on a standard Cartesian plane with the horizontal axis labeled as \( x \) and the vertical axis labeled as \( y \).
- **Range:** The \( x \)-axis ranges from \(-40\) to \(20\), and the \( y \)-axis ranges from \(-400\) to \(400\).
- **Graph Behavior:**
- The graph of the derivative function \( f'(x) \) dips sharply around \( x = 0 \).
- The curve then rises steeply after this dip.
- There is a visible local minimum close to the origin, indicating a shift in the curve's direction.
### Graphing Controls:
- The graph is implemented using Desmos, a graphing tool providing zoom in/out features, represented by plus (+) and minus (−) buttons.
- A home button is also present to reset the view.
### Description:
The function \( f'(x) = x^4 - 16x^3 + 91x^2 - 216x + 180 \) is a polynomial of degree 4, and its graph shows typical polynomial behavior with varying slopes and curvature. The graph is characterized by its steep decline and subsequent rise near the y-axis, indicative of changes in the slope and curvature of the original function.
![**Graph Analysis for Inflection Points and Concavity**
**Inflection Points:**
Use the graph to find the point(s) of inflection of \( f \).
(Give your answer as a comma-separated list of points in the form \((*, *)\). Use decimal notation. Give your numbers to three decimal places. Enter DNE if the function has no inflection points.)
- **inflection point(s):** [input box]
**Intervals of Concavity:**
Use the graph to determine the interval(s) on which the function \( f \) is concave down.
(Give your answer as an interval in the form \((*, *)\). Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and an appropriate type of parenthesis \("(", ")", "[", "]"\) depending on whether the interval is open or closed. Enter \(\emptyset\) if the interval is empty. Use decimal notation. Give your numbers to three decimal places.)
- **interval(s):** [input box]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bd5995a-c4b2-468d-a13c-4876a73e2cf6%2F17da13cd-6f70-490d-9de6-e7ab55e7b0f3%2Fqbxtl8_processed.png&w=3840&q=75)
Transcribed Image Text:**Graph Analysis for Inflection Points and Concavity**
**Inflection Points:**
Use the graph to find the point(s) of inflection of \( f \).
(Give your answer as a comma-separated list of points in the form \((*, *)\). Use decimal notation. Give your numbers to three decimal places. Enter DNE if the function has no inflection points.)
- **inflection point(s):** [input box]
**Intervals of Concavity:**
Use the graph to determine the interval(s) on which the function \( f \) is concave down.
(Give your answer as an interval in the form \((*, *)\). Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and an appropriate type of parenthesis \("(", ")", "[", "]"\) depending on whether the interval is open or closed. Enter \(\emptyset\) if the interval is empty. Use decimal notation. Give your numbers to three decimal places.)
- **interval(s):** [input box]
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