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(c) -4 -2 increasing decreasing 2 (i) Identify the critical numbers of f. (Enter your answers as a comma-separated list.) X = -1,0,1 (ii) Identify the open interval(s) on which f is increasing or decreasing. (Enter your answer using interval notation.) (-1/2) X - 12 ) ~ ( 1217,00) | -∞, - 4 X X Calculator

(c) -4 -2 increasing decreasing 2 (i) Identify the critical numbers of f. (Enter your answers as a comma-separated list.) X = -1,0,1 (ii) Identify the open interval(s) on which f is increasing or decreasing. (Enter your answer using interval notation.) (-1/2) X - 12 ) ~ ( 1217,00) | -∞, - 4 X X Calculator

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Graph Analysis and Questions #### Graph Description The graph displayed is of the derivative \( f' \) of a function \( f \). The x-axis ranges from -4 to 4, and the y-axis ranges from -4 to 2. There are significant changes in the slope, indicating critical points. #### Questions and Solutions (i) **Identify the critical numbers of \( f \).** - Enter your answers as a comma-separated list. - **Solution:** \( x = -1, 0, 1 \) (Correct) (ii) **Identify the open interval(s) on which \( f \) is increasing or decreasing.** - Enter your answer using interval notation. - **Increasing:** None of the given options are marked correct. - **Decreasing:** Neither of the given intervals are correct: - \(\left(-\infty, -\frac{1}{2}\right)\) - \(\left(\frac{1}{2}, \infty\right)\) (iii) **Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical number.** - Enter your answers as a comma-separated list. - **Relative Minimum:** - \( x = -\frac{1}{2} \) (Incorrect) - **Relative Maximum:** - \( x = \frac{1}{2} \) (Incorrect) ### Observations - The correct identification of critical numbers (where the derivative \( f' \) crosses the x-axis) is key to solving intervals and determining relative extrema. - Graph interpretation is necessary to accurately describe behavior such as increasing or decreasing intervals and the nature of critical points.
Transcribed Image Text:### Graph Analysis and Questions #### Graph Description The graph displayed is of the derivative \( f' \) of a function \( f \). The x-axis ranges from -4 to 4, and the y-axis ranges from -4 to 2. There are significant changes in the slope, indicating critical points. #### Questions and Solutions (i) **Identify the critical numbers of \( f \).** - Enter your answers as a comma-separated list. - **Solution:** \( x = -1, 0, 1 \) (Correct) (ii) **Identify the open interval(s) on which \( f \) is increasing or decreasing.** - Enter your answer using interval notation. - **Increasing:** None of the given options are marked correct. - **Decreasing:** Neither of the given intervals are correct: - \(\left(-\infty, -\frac{1}{2}\right)\) - \(\left(\frac{1}{2}, \infty\right)\) (iii) **Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical number.** - Enter your answers as a comma-separated list. - **Relative Minimum:** - \( x = -\frac{1}{2} \) (Incorrect) - **Relative Maximum:** - \( x = \frac{1}{2} \) (Incorrect) ### Observations - The correct identification of critical numbers (where the derivative \( f' \) crosses the x-axis) is key to solving intervals and determining relative extrema. - Graph interpretation is necessary to accurately describe behavior such as increasing or decreasing intervals and the nature of critical points.
### Graph Analysis of Derivatives The graph displays the derivative \( f' \) of a function \( f \). The graph is plotted on the Cartesian plane with the x-axis ranging from -4 to 4 and the y-axis ranging from -4 to 4. #### (i) Critical Numbers Identify the critical numbers of \( f \). - Answer: \( x = -1, 0, 1 \) #### (ii) Intervals of Increase and Decrease Determine the open interval(s) on which \( f \) is increasing or decreasing. Provide answers in interval notation. - Increasing: - Submitted Answer: \( \left( -\frac{1}{2}, \frac{1}{2} \right) \) (Incorrect) - Decreasing: - Submitted Answer: \( (-\infty, -\frac{1}{2}) \cup \left( \frac{1}{2}, \infty \right) \) (Incorrect) #### (iii) Relative Extrema Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical number. Provide answers as a comma-separated list. To properly analyze the intervals and extrema, one must understand that the graph \( f' \) represents the slope of the original function \( f \). Where \( f' \) is above the x-axis, \( f \) is increasing, and where \( f' \) is below the x-axis, \( f \) is decreasing. Critical numbers occur where \( f' = 0 \) or is undefined, indicating possible points of relative extrema.
Transcribed Image Text:### Graph Analysis of Derivatives The graph displays the derivative \( f' \) of a function \( f \). The graph is plotted on the Cartesian plane with the x-axis ranging from -4 to 4 and the y-axis ranging from -4 to 4. #### (i) Critical Numbers Identify the critical numbers of \( f \). - Answer: \( x = -1, 0, 1 \) #### (ii) Intervals of Increase and Decrease Determine the open interval(s) on which \( f \) is increasing or decreasing. Provide answers in interval notation. - Increasing: - Submitted Answer: \( \left( -\frac{1}{2}, \frac{1}{2} \right) \) (Incorrect) - Decreasing: - Submitted Answer: \( (-\infty, -\frac{1}{2}) \cup \left( \frac{1}{2}, \infty \right) \) (Incorrect) #### (iii) Relative Extrema Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical number. Provide answers as a comma-separated list. To properly analyze the intervals and extrema, one must understand that the graph \( f' \) represents the slope of the original function \( f \). Where \( f' \) is above the x-axis, \( f \) is increasing, and where \( f' \) is below the x-axis, \( f \) is decreasing. Critical numbers occur where \( f' = 0 \) or is undefined, indicating possible points of relative extrema.
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