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Consider the graph of g(x). -5 -4 -3 -2 O True 6 Ay 5 4 False an a - -2 -3 -4 1 2 3 y=g(x) True or False: If g(x) is the second derivative of f(x), then f(x) is concave down at = 2. 5

Consider the graph of g(x). -5 -4 -3 -2 O True 6 Ay 5 4 False an a - -2 -3 -4 1 2 3 y=g(x) True or False: If g(x) is the second derivative of f(x), then f(x) is concave down at = 2. 5

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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**Graph Analysis:** The image shows the graph of the function \( g(x) \), represented by a smooth, symmetric, downward-opening parabola. Here are the details of the graph: - **Axes:** The graph is plotted on the Cartesian coordinate plane with the horizontal axis labeled as \( x \) and the vertical axis labeled as \( y \). - **Equation:** The function is indicated by the equation \( y = g(x) \) shown in orange. - **Vertex:** The parabola’s peak, or vertex, is located at the point \((2, 6)\). - **Curve:** The parabola opens downward, extending from approximately \( x = -5 \) to \( x = 5 \) on the graph. **True or False Question:** The problem asks: "If \( g(x) \) is the second derivative of \( f(x) \), then \( f(x) \) is concave down at \( x = 2 \)." Options: - True - False **Explanation:** The second derivative, \( g(x) \), is positive to the left of \( x = 2 \) and negative to the right of \( x = 2 \). At \( x = 2 \), \( g(x) = 0 \), indicating a potential point of inflection for the function \( f(x) \). Since the graph is a downward-opening parabola, this suggests that \( f(x) \) is concave down at \( x = 2 \). Therefore, the statement is True.
Transcribed Image Text:**Graph Analysis:** The image shows the graph of the function \( g(x) \), represented by a smooth, symmetric, downward-opening parabola. Here are the details of the graph: - **Axes:** The graph is plotted on the Cartesian coordinate plane with the horizontal axis labeled as \( x \) and the vertical axis labeled as \( y \). - **Equation:** The function is indicated by the equation \( y = g(x) \) shown in orange. - **Vertex:** The parabola’s peak, or vertex, is located at the point \((2, 6)\). - **Curve:** The parabola opens downward, extending from approximately \( x = -5 \) to \( x = 5 \) on the graph. **True or False Question:** The problem asks: "If \( g(x) \) is the second derivative of \( f(x) \), then \( f(x) \) is concave down at \( x = 2 \)." Options: - True - False **Explanation:** The second derivative, \( g(x) \), is positive to the left of \( x = 2 \) and negative to the right of \( x = 2 \). At \( x = 2 \), \( g(x) = 0 \), indicating a potential point of inflection for the function \( f(x) \). Since the graph is a downward-opening parabola, this suggests that \( f(x) \) is concave down at \( x = 2 \). Therefore, the statement is True.
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A function is said to be concave down at a point where its second derivative is negative.

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