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Let f be a twice differentiable function on an open interval (a, b). Which statements regarding the second derivative and concavity are true? The concavity of a graph changes at an inflection point. If f is increasing, then the graph of f is concave down. If f" (c) is negative, then the graph of f has a local minimum at x = c. The graph of f is concave up if f" is positive on (a, b). The graph of f has a local maximum at x = c if f" (c) = 0.

Let f be a twice differentiable function on an open interval (a, b). Which statements regarding the second derivative and concavity are true? The concavity of a graph changes at an inflection point. If f is increasing, then the graph of f is concave down. If f" (c) is negative, then the graph of f has a local minimum at x = c. The graph of f is concave up if f" is positive on (a, b). The graph of f has a local maximum at x = c if f" (c) = 0.

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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### Understanding the Second Derivative and Concavity Let \( f \) be a twice differentiable function on an open interval \( (a, b) \). #### Which statements regarding the second derivative and concavity are true? - [ ] The concavity of a graph changes at an inflection point. - [ ] If \( f \) is increasing, then the graph of \( f \) is concave down. - [ ] If \( f''(c) \) is negative, then the graph of \( f \) has a local minimum at \( x = c \). - [ ] The graph of \( f \) is concave up if \( f'' \) is positive on \( (a, b) \). - [ ] The graph of \( f \) has a local maximum at \( x = c \) if \( f''(c) = 0 \). ### Explanation: 1. **Concavity and Inflection Points:** - The concavity of a graph changes at an inflection point. Inflection points are where the second derivative changes sign. 2. **Graph Increasing and Concavity:** - If a function is increasing (\( f' > 0 \)), this does not necessarily mean the graph is concave down. Concavity is determined by the second derivative, not the first. 3. **Second Derivative and Local Extrema:** - If the second derivative at \( c \) is negative (\( f''(c) < 0 \)), it actually indicates that the function has a local maximum at \( x = c \), not a local minimum. 4. **Positive Second Derivative and Concavity:** - The graph of a function is concave up on an interval if the second derivative is positive over that interval (\( f'' > 0 \)). 5. **Zero Second Derivative and Extrema:** - If the second derivative at \( c \) is zero (\( f''(c) = 0 \)), this indicates a possible inflection point, not necessarily a local maximum. Other criteria must be checked to determine if \( c \) is an inflection point or a local extremum. Understanding these principles is crucial when analyzing the behavior of functions regarding concavity and differentiability.
Transcribed Image Text:### Understanding the Second Derivative and Concavity Let \( f \) be a twice differentiable function on an open interval \( (a, b) \). #### Which statements regarding the second derivative and concavity are true? - [ ] The concavity of a graph changes at an inflection point. - [ ] If \( f \) is increasing, then the graph of \( f \) is concave down. - [ ] If \( f''(c) \) is negative, then the graph of \( f \) has a local minimum at \( x = c \). - [ ] The graph of \( f \) is concave up if \( f'' \) is positive on \( (a, b) \). - [ ] The graph of \( f \) has a local maximum at \( x = c \) if \( f''(c) = 0 \). ### Explanation: 1. **Concavity and Inflection Points:** - The concavity of a graph changes at an inflection point. Inflection points are where the second derivative changes sign. 2. **Graph Increasing and Concavity:** - If a function is increasing (\( f' > 0 \)), this does not necessarily mean the graph is concave down. Concavity is determined by the second derivative, not the first. 3. **Second Derivative and Local Extrema:** - If the second derivative at \( c \) is negative (\( f''(c) < 0 \)), it actually indicates that the function has a local maximum at \( x = c \), not a local minimum. 4. **Positive Second Derivative and Concavity:** - The graph of a function is concave up on an interval if the second derivative is positive over that interval (\( f'' > 0 \)). 5. **Zero Second Derivative and Extrema:** - If the second derivative at \( c \) is zero (\( f''(c) = 0 \)), this indicates a possible inflection point, not necessarily a local maximum. Other criteria must be checked to determine if \( c \) is an inflection point or a local extremum. Understanding these principles is crucial when analyzing the behavior of functions regarding concavity and differentiability.
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