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f(x) is continuous on (-∞0,00) and has critical numbers at x = a, b, c, and d. Use the sign chart for f'(x) to determine whether f has a local maximum, a local minimum, or neither at each critical number. f'(x) ND b Does f(x) have a local minimum, a local maximum, or no local extremum at x = a? Choose the correct answer below. OA. a local minimum O B. a local maximum OC. no local extremum Does f(x) have a local minimum, a local maximum, or no local extremum at x = b? Choose the correct answer below. OA. a local minimum OB. no local extremum OC. a local maximum Does f(x) have a local minimum, a local maximum, or no local extremum at x = c? Choose the correct answer below. OA. a local maximum OB. no local extremum OC. a local minimum

f(x) is continuous on (-∞0,00) and has critical numbers at x = a, b, c, and d. Use the sign chart for f'(x) to determine whether f has a local maximum, a local minimum, or neither at each critical number. f'(x) ND b Does f(x) have a local minimum, a local maximum, or no local extremum at x = a? Choose the correct answer below. OA. a local minimum O B. a local maximum OC. no local extremum Does f(x) have a local minimum, a local maximum, or no local extremum at x = b? Choose the correct answer below. OA. a local minimum OB. no local extremum OC. a local maximum Does f(x) have a local minimum, a local maximum, or no local extremum at x = c? Choose the correct answer below. OA. a local maximum OB. no local extremum OC. a local minimum

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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### Analysis of Local Extrema Using the Sign Chart for \( f'(x) \) Consider a function \( f(x) \) that is continuous on \( (-\infty, \infty) \) and has critical numbers at \( x = a, b, c, \) and \( d \). The objective is to use the sign chart for \( f'(x) \) to determine whether \( f \) has a local maximum, a local minimum, or neither at each critical number. #### Sign Chart for \( f'(x) \): \[ \begin{array}{ccccccccc} & + & + & 0 & - & - & - & 0 & + & 0 & - & - & 0 & . . . \end{array} \] \[ \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a \ \ \ \ \ \ \ \ \ \ \ \ \ b \ \ \ \ \ \ \ \ \ \ \ \ \ c \ \ \ \ \ \ \ \ \ \ \ \ \ d \ \ \ \ \ } \] ### Questions: 1. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = a \)? Choose the correct answer below.** - A. a local minimum - B. a local maximum - C. no local extremum 2. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = b \)? Choose the correct answer below.** - A. a local minimum - B. no local extremum - C. a local maximum 3. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = c \)? Choose the correct answer below.** - A. a local maximum - B. no local extremum - C. a local minimum ### Explanation of the Sign Chart: The sign chart for \( f'(x) \) indicates the intervals where the derivative of the function \( f(x) \) is positive or negative. - When \( f'(x) \) transitions from positive to zero to negative, it suggests a local
Transcribed Image Text:### Analysis of Local Extrema Using the Sign Chart for \( f'(x) \) Consider a function \( f(x) \) that is continuous on \( (-\infty, \infty) \) and has critical numbers at \( x = a, b, c, \) and \( d \). The objective is to use the sign chart for \( f'(x) \) to determine whether \( f \) has a local maximum, a local minimum, or neither at each critical number. #### Sign Chart for \( f'(x) \): \[ \begin{array}{ccccccccc} & + & + & 0 & - & - & - & 0 & + & 0 & - & - & 0 & . . . \end{array} \] \[ \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a \ \ \ \ \ \ \ \ \ \ \ \ \ b \ \ \ \ \ \ \ \ \ \ \ \ \ c \ \ \ \ \ \ \ \ \ \ \ \ \ d \ \ \ \ \ } \] ### Questions: 1. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = a \)? Choose the correct answer below.** - A. a local minimum - B. a local maximum - C. no local extremum 2. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = b \)? Choose the correct answer below.** - A. a local minimum - B. no local extremum - C. a local maximum 3. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = c \)? Choose the correct answer below.** - A. a local maximum - B. no local extremum - C. a local minimum ### Explanation of the Sign Chart: The sign chart for \( f'(x) \) indicates the intervals where the derivative of the function \( f(x) \) is positive or negative. - When \( f'(x) \) transitions from positive to zero to negative, it suggests a local
### Understanding Local Extrema Using the First Derivative Test **Objective:** Determine whether the function \( f \) has a local maximum, a local minimum, or neither at each critical number using the first derivative test. **Given:** \( f(x) \) is continuous on \( (-\infty, \infty) \) and has critical numbers at \( x = a, b, c, \) and \( d \). Utilize the sign chart for \( f'(x) \) to analyze the behavior of \( f \). **Sign Chart:** ``` (+) (-) (+) (-) (+) (0) --------- a -------- b ----- c -------- d ------> f'(x) ``` **Analysis:** 1. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = b \)?** Choose the correct answer below. A. ○ local minimum B. ○ no local extremum C. ● local maximum 2. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = c \)?** Choose the correct answer below. A. ● local maximum B. ○ no local extremum C. ○ local minimum 3. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x=d \)?** Choose the correct answer below. A. ● local maximum B. ○ local minimum C. ○ no local extremum **Explanation of the First Derivative Test:** The first derivative test is used to determine if a given point is a local maximum, minimum, or neither by analyzing the sign changes of the derivative \( f'(x) \). - **Local Maximum:** If \( f'(x) \) changes from positive to negative at a critical point. - **Local Minimum:** If \( f'(x) \) changes from negative to positive at a critical point. - **No Local Extrema:** If there is no sign change at the critical point. **Sign Chart Details:** - **At \( x=a \):** \( f'(x) \) changes from positive to negative, indicating a local maximum. - **At \( x=b \):** \( f
Transcribed Image Text:### Understanding Local Extrema Using the First Derivative Test **Objective:** Determine whether the function \( f \) has a local maximum, a local minimum, or neither at each critical number using the first derivative test. **Given:** \( f(x) \) is continuous on \( (-\infty, \infty) \) and has critical numbers at \( x = a, b, c, \) and \( d \). Utilize the sign chart for \( f'(x) \) to analyze the behavior of \( f \). **Sign Chart:** ``` (+) (-) (+) (-) (+) (0) --------- a -------- b ----- c -------- d ------> f'(x) ``` **Analysis:** 1. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = b \)?** Choose the correct answer below. A. ○ local minimum B. ○ no local extremum C. ● local maximum 2. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x = c \)?** Choose the correct answer below. A. ● local maximum B. ○ no local extremum C. ○ local minimum 3. **Does \( f(x) \) have a local minimum, a local maximum, or no local extremum at \( x=d \)?** Choose the correct answer below. A. ● local maximum B. ○ local minimum C. ○ no local extremum **Explanation of the First Derivative Test:** The first derivative test is used to determine if a given point is a local maximum, minimum, or neither by analyzing the sign changes of the derivative \( f'(x) \). - **Local Maximum:** If \( f'(x) \) changes from positive to negative at a critical point. - **Local Minimum:** If \( f'(x) \) changes from negative to positive at a critical point. - **No Local Extrema:** If there is no sign change at the critical point. **Sign Chart Details:** - **At \( x=a \):** \( f'(x) \) changes from positive to negative, indicating a local maximum. - **At \( x=b \):** \( f
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