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2) Draw a graph of a continuous function that satisfies the following conditions. f'(4) > 0 f'(-6) < 0 f" (2) < 0 f(-1) = 3

2) Draw a graph of a continuous function that satisfies the following conditions. f'(4) > 0 f'(-6) < 0 f" (2) < 0 f(-1) = 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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### Problem Description **Task:** Draw a graph of a continuous function that satisfies the following conditions: 1. \( f'(4) > 0 \) 2. \( f'(-6) < 0 \) 3. \( f''(2) < 0 \) 4. \( f(-1) = 3 \) ### Explanation of the Graph #### Axes and Scale - The graph is drawn on a standard Cartesian coordinate system. - The x-axis is labeled and marked with integers ranging from -8 to 8. - The y-axis extends both upwards and downwards from the origin. #### Conditions Explained 1. **\( f'(4) > 0 \):** The slope of the function at \( x = 4 \) is positive. This implies the graph is increasing at this point. 2. **\( f'(-6) < 0 \):** The slope of the function at \( x = -6 \) is negative. This implies the graph is decreasing at this point. 3. **\( f''(2) < 0 \):** The second derivative at \( x = 2 \) is negative, indicating that the graph is concave down at this point. 4. **\( f(-1) = 3 \):** The value of the function at \( x = -1 \) is 3. This means that the point \((-1, 3)\) lies on the graph. This exercise involves interpreting and sketching a graph based on derivative conditions, showcasing how the first and second derivatives affect the shape of the function.
Transcribed Image Text:### Problem Description **Task:** Draw a graph of a continuous function that satisfies the following conditions: 1. \( f'(4) > 0 \) 2. \( f'(-6) < 0 \) 3. \( f''(2) < 0 \) 4. \( f(-1) = 3 \) ### Explanation of the Graph #### Axes and Scale - The graph is drawn on a standard Cartesian coordinate system. - The x-axis is labeled and marked with integers ranging from -8 to 8. - The y-axis extends both upwards and downwards from the origin. #### Conditions Explained 1. **\( f'(4) > 0 \):** The slope of the function at \( x = 4 \) is positive. This implies the graph is increasing at this point. 2. **\( f'(-6) < 0 \):** The slope of the function at \( x = -6 \) is negative. This implies the graph is decreasing at this point. 3. **\( f''(2) < 0 \):** The second derivative at \( x = 2 \) is negative, indicating that the graph is concave down at this point. 4. **\( f(-1) = 3 \):** The value of the function at \( x = -1 \) is 3. This means that the point \((-1, 3)\) lies on the graph. This exercise involves interpreting and sketching a graph based on derivative conditions, showcasing how the first and second derivatives affect the shape of the function.
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