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Select a possible graph of y = f(x), using the given information about the derivatives y' = f'(x) and y" = ƒ" (x). Assume that the function is defined and continuous for all real x. y = 0 y <0! X1 1 y' =0 y">0 y < 0 y"' < 0 x2 i y' = 0 y">0 X x

Select a possible graph of y = f(x), using the given information about the derivatives y' = f'(x) and y" = ƒ" (x). Assume that the function is defined and continuous for all real x. y = 0 y <0! X1 1 y' =0 y">0 y < 0 y"' < 0 x2 i y' = 0 y">0 X x

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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**Explanation of the Graph:** The image presents a guide to selecting a possible graph of \( y = f(x) \) by utilizing information about its first and second derivatives, \( y' = f'(x) \) and \( y'' = f''(x) \). The function is defined and continuous for all real \( x \). **Graph Details:** 1. **Horizontal Axis (x-axis):** - Represents the range of \( x \) values. 2. **Vertical Axis (not explicitly shown):** - Represents the behavior of the function \( y = f(x) \). 3. **Critical Points:** - Two critical points are shown at \( x_1 \) and \( x_2 \). 4. **First Derivative (\( y' \)):** - At \( x_1 \) and \( x_2 \), \( y' = 0 \) indicating potential local maxima, minima, or inflection points. - For \( x < x_1 \), \( y' < 0 \), meaning the function is decreasing. - For \( x_1 < x < x_2 \), \( y' < 0 \), meaning the function continues to decrease. - For \( x > x_2 \), \( y' > 0 \), meaning the function starts increasing. 5. **Second Derivative (\( y'' \)):** - At \( x_1 \) and \( x_2 \), \( y'' = 0 \) indicating possible points of inflection. - For \( x < x_1 \), \( y'' > 0 \), indicating the function is concave up. - For \( x_1 < x < x_2 \), \( y'' < 0 \), indicating the function is concave down. - For \( x > x_2 \), \( y'' > 0 \), indicating the function is concave up. This information helps in sketching the potential shape of the function \( y = f(x) \), illustrating the behavior around critical points and intervals of concavity.
Transcribed Image Text:**Explanation of the Graph:** The image presents a guide to selecting a possible graph of \( y = f(x) \) by utilizing information about its first and second derivatives, \( y' = f'(x) \) and \( y'' = f''(x) \). The function is defined and continuous for all real \( x \). **Graph Details:** 1. **Horizontal Axis (x-axis):** - Represents the range of \( x \) values. 2. **Vertical Axis (not explicitly shown):** - Represents the behavior of the function \( y = f(x) \). 3. **Critical Points:** - Two critical points are shown at \( x_1 \) and \( x_2 \). 4. **First Derivative (\( y' \)):** - At \( x_1 \) and \( x_2 \), \( y' = 0 \) indicating potential local maxima, minima, or inflection points. - For \( x < x_1 \), \( y' < 0 \), meaning the function is decreasing. - For \( x_1 < x < x_2 \), \( y' < 0 \), meaning the function continues to decrease. - For \( x > x_2 \), \( y' > 0 \), meaning the function starts increasing. 5. **Second Derivative (\( y'' \)):** - At \( x_1 \) and \( x_2 \), \( y'' = 0 \) indicating possible points of inflection. - For \( x < x_1 \), \( y'' > 0 \), indicating the function is concave up. - For \( x_1 < x < x_2 \), \( y'' < 0 \), indicating the function is concave down. - For \( x > x_2 \), \( y'' > 0 \), indicating the function is concave up. This information helps in sketching the potential shape of the function \( y = f(x) \), illustrating the behavior around critical points and intervals of concavity.
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