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List the open interval(s) on which f

List the open interval(s) on which f

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Interactive Graph and Problem Set: Understanding Concavity and Inflection Points** **Graph Description:** The graph featured above illustrates a function \( f(x) \) plotted over an interval on the x-axis. Observing the graph, the function displays sections that bend either upwards or downwards and crosses the x-axis at least once. Such behavior will guide the analysis of concavity. **Problem Set:** 1. **Concave Upward Intervals:** - *Task:* List the open interval(s) on which the function \( f \) is concave upward. Ensure that your intervals are as large as possible. - *Answer (in interval notation):* \( (0, 2) \cup (5, 8) \) 2. **Concave Downward Intervals:** - *Task:* List the open interval(s) on which the function \( f \) is concave downward. Again, make sure your intervals are maximized. - *Answer (in interval notation):* \( (2, 4) \cup (4, 5) \) 3. **Points of Inflection:** - *Task:* Identify the points of inflection where the function changes concavity. Provide your answers as points in the form \((a, b)\). - *Answer (separated by commas):* \( (2, 5), (5, 5) \) **Note:** Clicking on the graph allows for an enlarged view, providing a clearer visualization of the concavity changes and inflection points.
Transcribed Image Text:**Interactive Graph and Problem Set: Understanding Concavity and Inflection Points** **Graph Description:** The graph featured above illustrates a function \( f(x) \) plotted over an interval on the x-axis. Observing the graph, the function displays sections that bend either upwards or downwards and crosses the x-axis at least once. Such behavior will guide the analysis of concavity. **Problem Set:** 1. **Concave Upward Intervals:** - *Task:* List the open interval(s) on which the function \( f \) is concave upward. Ensure that your intervals are as large as possible. - *Answer (in interval notation):* \( (0, 2) \cup (5, 8) \) 2. **Concave Downward Intervals:** - *Task:* List the open interval(s) on which the function \( f \) is concave downward. Again, make sure your intervals are maximized. - *Answer (in interval notation):* \( (2, 4) \cup (4, 5) \) 3. **Points of Inflection:** - *Task:* Identify the points of inflection where the function changes concavity. Provide your answers as points in the form \((a, b)\). - *Answer (separated by commas):* \( (2, 5), (5, 5) \) **Note:** Clicking on the graph allows for an enlarged view, providing a clearer visualization of the concavity changes and inflection points.
### Educational Exercise: Analyzing the Graph of a Function **Instructions:** Use the given graph of the function \( f \) to answer the following questions. If a question has two or more answers, separate them with commas. **Graph Description:** The graph depicted is a smooth, continuous curve of a mathematical function \( f \). The curve initiates on the left at a y-value above the x-axis, descends steeply below the x-axis, rises back above it, and displays a wavy pattern with multiple peaks and troughs. - Axis Labels: The x-axis ranges from approximately -1 to 8. The y-axis is marked from -1 to above 1. - Key features include: - A local minimum at approximately x = 0.5. - A local maximum near x = 2.5. - A local minimum around x = 5. - The curve becomes steeper after x = 6, continuing upward. **Questions:** 1. **List the open interval(s) on which \( f \) is concave upward.** - For credit, your interval(s) must be as large as possible. - **Answer (in interval notation):** \((0, 2) \cup (5, 8)\) 2. **List the open interval(s) on which \( f \) is concave downward.** - (Answer the same way as above.) ### Note: Concavity refers to the orientation of the graph. A graph is concave upward on an interval if it lies above its tangent lines and concave downward if it lies below its tangent lines.
Transcribed Image Text:### Educational Exercise: Analyzing the Graph of a Function **Instructions:** Use the given graph of the function \( f \) to answer the following questions. If a question has two or more answers, separate them with commas. **Graph Description:** The graph depicted is a smooth, continuous curve of a mathematical function \( f \). The curve initiates on the left at a y-value above the x-axis, descends steeply below the x-axis, rises back above it, and displays a wavy pattern with multiple peaks and troughs. - Axis Labels: The x-axis ranges from approximately -1 to 8. The y-axis is marked from -1 to above 1. - Key features include: - A local minimum at approximately x = 0.5. - A local maximum near x = 2.5. - A local minimum around x = 5. - The curve becomes steeper after x = 6, continuing upward. **Questions:** 1. **List the open interval(s) on which \( f \) is concave upward.** - For credit, your interval(s) must be as large as possible. - **Answer (in interval notation):** \((0, 2) \cup (5, 8)\) 2. **List the open interval(s) on which \( f \) is concave downward.** - (Answer the same way as above.) ### Note: Concavity refers to the orientation of the graph. A graph is concave upward on an interval if it lies above its tangent lines and concave downward if it lies below its tangent lines.
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