**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]

Given: f'x=x4-16x3+91x2-216x+180 To Find: i. Inflection Point(ii). Interval of concavity Concept:…

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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Question

**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]

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