Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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## Educational Website Content: Understanding Graphs of Second Derivatives

### Graph Analysis

The graph displayed represents the second derivative of a function, denoted as \( f''(x) \).

#### Graph Description

- **Axes:**
  - The horizontal axis is labeled \( x \).
  - The vertical axis is labeled \( y \), with values ranging from -1000 to 1000.
  
- **Curve Behavior:**
  - The graph shows a smooth curve that intersects the \( x \)-axis at three points: \( x = -4 \), \( x = 0 \), and \( x = 4 \).
  - The curve exhibits positive and negative values, indicating intervals of concavity and convexity for the original function \( f(x) \).

### Inflection Points

An inflection point occurs where the concavity of \( f(x) \) changes, which corresponds to the points where the second derivative \( f''(x) = 0 \).

#### Exercise

1. **Determine the Number of Inflection Points:**
   - Based on the graph, identify how many times \( f''(x) \) equals zero.

2. **X-Coordinates of Inflection Points:**
   - Enter the \( x \)-coordinates as a comma-separated list where \( f''(x) \) intersects the \( x \)-axis.

### Instructions

- Enter the number of inflection points in the first box.
- Provide the \( x \)-coordinates for the points of inflection in the second box.
- If \( f(x) \) does not have any inflection points, input "DNE" (Does Not Exist).

*Note:* Ensure the graph is properly loaded if not visible. This helps in analyzing the behavior of the function for further calculus studies.
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Transcribed Image Text:## Educational Website Content: Understanding Graphs of Second Derivatives ### Graph Analysis The graph displayed represents the second derivative of a function, denoted as \( f''(x) \). #### Graph Description - **Axes:** - The horizontal axis is labeled \( x \). - The vertical axis is labeled \( y \), with values ranging from -1000 to 1000. - **Curve Behavior:** - The graph shows a smooth curve that intersects the \( x \)-axis at three points: \( x = -4 \), \( x = 0 \), and \( x = 4 \). - The curve exhibits positive and negative values, indicating intervals of concavity and convexity for the original function \( f(x) \). ### Inflection Points An inflection point occurs where the concavity of \( f(x) \) changes, which corresponds to the points where the second derivative \( f''(x) = 0 \). #### Exercise 1. **Determine the Number of Inflection Points:** - Based on the graph, identify how many times \( f''(x) \) equals zero. 2. **X-Coordinates of Inflection Points:** - Enter the \( x \)-coordinates as a comma-separated list where \( f''(x) \) intersects the \( x \)-axis. ### Instructions - Enter the number of inflection points in the first box. - Provide the \( x \)-coordinates for the points of inflection in the second box. - If \( f(x) \) does not have any inflection points, input "DNE" (Does Not Exist). *Note:* Ensure the graph is properly loaded if not visible. This helps in analyzing the behavior of the function for further calculus studies.
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