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Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(2) = f'(4) = 0, f'(x) > 0 if x < 0 or 2 < x < 4, f'(x) < 0 if 0 < x < 2 or x > 4, f"(x) > 0 if 1 < x < 3, f"(x) < 0 if x < 1 or x > 3

Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(2) = f'(4) = 0, f'(x) > 0 if x < 0 or 2 < x < 4, f'(x) < 0 if 0 < x < 2 or x > 4, f"(x) > 0 if 1 < x < 3, f"(x) < 0 if x < 1 or x > 3

Calculus: Early Transcendentals
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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**Assignment: Sketch the Graph of a Function** **Objective:** Create a graph of a function that meets all the given criteria. **Conditions:** - \( f'(0) = f'(2) = f'(4) = 0 \) - \( f'(x) > 0 \) for \( x < 0 \) or \( 2 < x < 4 \) - \( f'(x) < 0 \) for \( 0 < x < 2 \) or \( x > 4 \) - \( f''(x) > 0 \) for \( 1 < x < 3 \) - \( f''(x) < 0 \) for \( x < 1 \) or \( x > 3 \) **Diagrams:** 1. **Top Left Graph:** - The function starts increasing before \( x = 0 \). - Decreases between \( 0 < x < 2 \). - Increases between \( 2 < x < 4 \). - Decreases after \( x > 4 \). - Inflection points observed at \( x = 1 \) and \( x = 3 \). 2. **Top Right Graph:** - Similar increases and decreases but with a different amplitude. - Function behavior is consistent with \( f'(x) \) and \( f''(x) \) conditions but appears incorrectly scaled. 3. **Bottom Left Graph:** - Shows wavelike pattern. - Increases and decreases are subtler, making it less visibly fitting the criteria. 4. **Bottom Right Graph:** - Starts increasing, peaks near \( x = 0 \), decreases and has a visible curve between 1 and 3 suggesting \( f''(x) > 0 \). - Incorrect scaling or detail to fully satisfy all conditions. **Task:** Choose the graph that best meets all conditions, noting changes at each critical point, ensuring all derivatives conform to specified behavior.
Transcribed Image Text:**Assignment: Sketch the Graph of a Function** **Objective:** Create a graph of a function that meets all the given criteria. **Conditions:** - \( f'(0) = f'(2) = f'(4) = 0 \) - \( f'(x) > 0 \) for \( x < 0 \) or \( 2 < x < 4 \) - \( f'(x) < 0 \) for \( 0 < x < 2 \) or \( x > 4 \) - \( f''(x) > 0 \) for \( 1 < x < 3 \) - \( f''(x) < 0 \) for \( x < 1 \) or \( x > 3 \) **Diagrams:** 1. **Top Left Graph:** - The function starts increasing before \( x = 0 \). - Decreases between \( 0 < x < 2 \). - Increases between \( 2 < x < 4 \). - Decreases after \( x > 4 \). - Inflection points observed at \( x = 1 \) and \( x = 3 \). 2. **Top Right Graph:** - Similar increases and decreases but with a different amplitude. - Function behavior is consistent with \( f'(x) \) and \( f''(x) \) conditions but appears incorrectly scaled. 3. **Bottom Left Graph:** - Shows wavelike pattern. - Increases and decreases are subtler, making it less visibly fitting the criteria. 4. **Bottom Right Graph:** - Starts increasing, peaks near \( x = 0 \), decreases and has a visible curve between 1 and 3 suggesting \( f''(x) > 0 \). - Incorrect scaling or detail to fully satisfy all conditions. **Task:** Choose the graph that best meets all conditions, noting changes at each critical point, ensuring all derivatives conform to specified behavior.
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