Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
### Critical Points and Inflection Points In this exercise, you are asked to identify critical points and inflection points of a given function based on its derivative conditions. #### Part (a): Finding Critical Points A critical point for a function \( f(x) \) is where its first derivative \( f'(x) \) equals zero. These points can indicate local maxima, minima, or saddle points. **Question:** Identify the values of \( x \) at which the first derivative \( f'(x) \) equals zero. \[ f'(x) = 0 \] Options: - [ ] \( x_0 \) - [ ] \( x_1 \) - [ ] \( x_2 \) - [ ] \( x_3 \) - [ ] \( x_4 \) - [ ] none of these #### Part (b): Finding Inflection Points An inflection point for a function \( f(x) \) is where its second derivative \( f''(x) \) equals zero and the concavity changes. **Question:** Identify the values of \( x \) at which the second derivative \( f''(x) \) equals zero. \[ f''(x) = 0 \] Options: - [ ] \( x_0 \) - [ ] \( x_1 \) - [ ] \( x_2 \) - [ ] \( x_3 \) - [ ] \( x_4 \) - [ ] none of these Select the appropriate options to indicate the values where \( f'(x) = 0 \) and \( f''(x) = 0 \).
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Transcribed Image Text:### Critical Points and Inflection Points In this exercise, you are asked to identify critical points and inflection points of a given function based on its derivative conditions. #### Part (a): Finding Critical Points A critical point for a function \( f(x) \) is where its first derivative \( f'(x) \) equals zero. These points can indicate local maxima, minima, or saddle points. **Question:** Identify the values of \( x \) at which the first derivative \( f'(x) \) equals zero. \[ f'(x) = 0 \] Options: - [ ] \( x_0 \) - [ ] \( x_1 \) - [ ] \( x_2 \) - [ ] \( x_3 \) - [ ] \( x_4 \) - [ ] none of these #### Part (b): Finding Inflection Points An inflection point for a function \( f(x) \) is where its second derivative \( f''(x) \) equals zero and the concavity changes. **Question:** Identify the values of \( x \) at which the second derivative \( f''(x) \) equals zero. \[ f''(x) = 0 \] Options: - [ ] \( x_0 \) - [ ] \( x_1 \) - [ ] \( x_2 \) - [ ] \( x_3 \) - [ ] \( x_4 \) - [ ] none of these Select the appropriate options to indicate the values where \( f'(x) = 0 \) and \( f''(x) = 0 \).
### Identifying Critical Points on a Function Graph #### Problem Statement Identify the real numbers \( x_0, x_1, x_2, x_3 \), and \( x_4 \) in the figure such that each of the following statements is true. (Select all that apply). #### Graph Description The graph depicts a function \( f \) plotted against the x and y axes. The graph shows a continuous curve that varies with peaks and troughs. Key points on the x-axis are marked as \( x_0, x_1, x_2, x_3 \), and \( x_4 \). Each of these points corresponds to significant features of the function \( f \). 1. **Axes**: - **x-axis**: A horizontal line marking the real numbers. - **y-axis**: A vertical line marking the function values. 2. **Curve Description**: - The curve starts high on the left, descends to a local minimum at \( x_0 \), ascends to a local maximum at \( x_1 \), descends again to a local minimum at \( x_2 \), rises to a local maximum at \( x_3 \), and then descends slightly to a local minimum approaching \( x_4 \). 3. **Vertical Dotted Lines**: - Dotted lines are drawn from each marked x-value on the x-axis (\( x_0, x_1, x_2, x_3, \) and \( x_4 \)) to their respective points on the curve. This graphical representation helps in understanding the critical points of the function, namely local minima and maxima, which are essential in various fields of study, including calculus and optimization. Use the visual cues from the graph to identify the nature of these critical points and solve related problems effectively.
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Transcribed Image Text:### Identifying Critical Points on a Function Graph #### Problem Statement Identify the real numbers \( x_0, x_1, x_2, x_3 \), and \( x_4 \) in the figure such that each of the following statements is true. (Select all that apply). #### Graph Description The graph depicts a function \( f \) plotted against the x and y axes. The graph shows a continuous curve that varies with peaks and troughs. Key points on the x-axis are marked as \( x_0, x_1, x_2, x_3 \), and \( x_4 \). Each of these points corresponds to significant features of the function \( f \). 1. **Axes**: - **x-axis**: A horizontal line marking the real numbers. - **y-axis**: A vertical line marking the function values. 2. **Curve Description**: - The curve starts high on the left, descends to a local minimum at \( x_0 \), ascends to a local maximum at \( x_1 \), descends again to a local minimum at \( x_2 \), rises to a local maximum at \( x_3 \), and then descends slightly to a local minimum approaching \( x_4 \). 3. **Vertical Dotted Lines**: - Dotted lines are drawn from each marked x-value on the x-axis (\( x_0, x_1, x_2, x_3, \) and \( x_4 \)) to their respective points on the curve. This graphical representation helps in understanding the critical points of the function, namely local minima and maxima, which are essential in various fields of study, including calculus and optimization. Use the visual cues from the graph to identify the nature of these critical points and solve related problems effectively.
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