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-6 -4 X = -2 y decreasing 6 4 -2 -4 -6 2 4 (i) Identify the critical numbers of f. (Enter your answers as a comma-separated list.) 6 relative maximum X = X (ii) Identify the open interval(s) on which f is increasing or decreasing. (Enter your answer usin notation.) increasing (iii) Determine whether f has a relative maximum, a relative minimum, or neither at each critic (Enter your answers as a comma-separated list.) relative minimum X =

-6 -4 X = -2 y decreasing 6 4 -2 -4 -6 2 4 (i) Identify the critical numbers of f. (Enter your answers as a comma-separated list.) 6 relative maximum X = X (ii) Identify the open interval(s) on which f is increasing or decreasing. (Enter your answer usin notation.) increasing (iii) Determine whether f has a relative maximum, a relative minimum, or neither at each critic (Enter your answers as a comma-separated list.) relative minimum X =

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: Derivative and Critical Points The graph displayed represents the derivative \( f' \) of a function \( f \) over the interval \([-6, 6]\) on the x-axis and [-6, 6] on the y-axis. #### Key Features of the Graph: - The graph crosses the x-axis at approximately \( x = -4 \), \( x = -1 \), and \( x = 3 \). - The function \( f' \) changes from positive to negative at \( x = -4 \) and \( x = 3 \), suggesting these are critical points where local maxima or minima of \( f \) might occur. - The graph \( f' \) transitions from negative to positive at \( x = -1 \), suggesting a critical point for a local minimum of \( f \). ### Questions: (i) **Identify the critical numbers of \( f \).** _Enter your answers as a comma-separated list._ \[ x = \_\_\_\_ \] (ii) **Identify the open interval(s) on which \( f \) is increasing or decreasing.** _Enter your answer using interval notation._ - **Increasing:** \(\_\_\_\_\) - **Decreasing:** \(\_\_\_\_\) (iii) **Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical point.** _Enter your answers as a comma-separated list._ - **Relative minimum:** \( x = \_\_\_\_ \) - **Relative maximum:** \( x = \_\_\_\_ \) This exercise helps in understanding the relationship between the derivative of a function and its critical points, intervals of increase and decrease, and identifying local maxima and minima.
Transcribed Image Text:### Graph Analysis: Derivative and Critical Points The graph displayed represents the derivative \( f' \) of a function \( f \) over the interval \([-6, 6]\) on the x-axis and [-6, 6] on the y-axis. #### Key Features of the Graph: - The graph crosses the x-axis at approximately \( x = -4 \), \( x = -1 \), and \( x = 3 \). - The function \( f' \) changes from positive to negative at \( x = -4 \) and \( x = 3 \), suggesting these are critical points where local maxima or minima of \( f \) might occur. - The graph \( f' \) transitions from negative to positive at \( x = -1 \), suggesting a critical point for a local minimum of \( f \). ### Questions: (i) **Identify the critical numbers of \( f \).** _Enter your answers as a comma-separated list._ \[ x = \_\_\_\_ \] (ii) **Identify the open interval(s) on which \( f \) is increasing or decreasing.** _Enter your answer using interval notation._ - **Increasing:** \(\_\_\_\_\) - **Decreasing:** \(\_\_\_\_\) (iii) **Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical point.** _Enter your answers as a comma-separated list._ - **Relative minimum:** \( x = \_\_\_\_ \) - **Relative maximum:** \( x = \_\_\_\_ \) This exercise helps in understanding the relationship between the derivative of a function and its critical points, intervals of increase and decrease, and identifying local maxima and minima.
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