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Use the graph of f '(x) shown below to answer each of the following. Note that the domain of f(z) is (-0, c) U (c, 0). at

Use the graph of f '(x) shown below to answer each of the following. Note that the domain of f(z) is (-0, c) U (c, 0). at

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 Statement:** iv) Find the interval(s) where \( f(x) \) is concave down. v) Find the \( x \)-values where any inflection points of \( f(x) \) occur. **Explanation:** - **Concavity and Intervals:** In calculus, a function is said to be concave down on an interval if its second derivative is negative over that interval. This indicates that the graph of the function is bending downwards. - **Inflection Points:** An inflection point occurs where the concavity of a function changes. This means the second derivative will change signs at these points. To solve these problems, you should: 1. Calculate the second derivative of \( f(x) \). 2. Determine where the second derivative is negative to find concave down intervals. 3. Identify the \( x \)-values where the second derivative changes from positive to negative or vice versa to find inflection points.
Transcribed Image Text:**Problem Statement:** iv) Find the interval(s) where \( f(x) \) is concave down. v) Find the \( x \)-values where any inflection points of \( f(x) \) occur. **Explanation:** - **Concavity and Intervals:** In calculus, a function is said to be concave down on an interval if its second derivative is negative over that interval. This indicates that the graph of the function is bending downwards. - **Inflection Points:** An inflection point occurs where the concavity of a function changes. This means the second derivative will change signs at these points. To solve these problems, you should: 1. Calculate the second derivative of \( f(x) \). 2. Determine where the second derivative is negative to find concave down intervals. 3. Identify the \( x \)-values where the second derivative changes from positive to negative or vice versa to find inflection points.
**Text:** Use the graph of \( f'(x) \) shown below to answer each of the following. Note that the domain of \( f(x) \) is \((-\infty, c) \cup (c, \infty)\). **Graph Explanation:** The graph depicted is of the derivative \( f'(x) \). It features several key elements: - The horizontal axis is labeled \( x \), and the vertical axis is labeled \( f'(x) \). - The graph passes through the point labeled \( a \) and reaches a minimum at the origin \( 0 \). - The curve rises, passing through a vertical asymptote at \( x = c \), creating a discontinuity. - There are open circles at points \( b \) and \( e \), indicating undefined values of the derivative at these points. - The curve approaches a vertical asymptote between points \( d \) and continues into positive values as \( x \) increases. - The function \( f'(x) \) is negative between points \( a \) and \( d \), and positive after point \( d \).
Transcribed Image Text:**Text:** Use the graph of \( f'(x) \) shown below to answer each of the following. Note that the domain of \( f(x) \) is \((-\infty, c) \cup (c, \infty)\). **Graph Explanation:** The graph depicted is of the derivative \( f'(x) \). It features several key elements: - The horizontal axis is labeled \( x \), and the vertical axis is labeled \( f'(x) \). - The graph passes through the point labeled \( a \) and reaches a minimum at the origin \( 0 \). - The curve rises, passing through a vertical asymptote at \( x = c \), creating a discontinuity. - There are open circles at points \( b \) and \( e \), indicating undefined values of the derivative at these points. - The curve approaches a vertical asymptote between points \( d \) and continues into positive values as \( x \) increases. - The function \( f'(x) \) is negative between points \( a \) and \( d \), and positive after point \( d \).
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