Use the graph of f' to identify the critical numbers of f, identify the open intervals on which f is increasing or decreasing, and determine whether f had a relative maximum, a relative minimum or neither at each critical number.
Use the graph of f' to identify the critical numbers of f, identify the open intervals on which f is increasing or decreasing, and determine whether f had a relative maximum, a relative minimum or neither at each critical number.
Use the graph of f' to identify the critical numbers of f, identify the open intervals on which f is increasing or decreasing, and determine whether f had a relative maximum, a relative minimum or neither at each critical number.
Use the graph of f' to identify the critical numbers of f, identify the open intervals on which f is increasing or decreasing, and determine whether f had a relative maximum, a relative minimum or neither at each critical number.
Transcribed Image Text:### Analyzing a Function's Derivative for Critical Points, Intervals, and Extrema
Consider the graph of the derivative \( f' \) of a function \( f \) shown below. The graph will assist in identifying the critical numbers, intervals where the function is increasing or decreasing, and determining relative minima and maxima of \( f \).
#### Graph Description:
- The graph displays the derivative \( f' \) with the \( y \)-axis ranging from \(-6\) to \(6\) and the \( x \)-axis ranging from \(-6\) to \(6\).
- The curve intersects the \( x \)-axis at \( x = -2 \), \( x = 0 \), and \( x = 3 \).
- Key features of the graph include:
- A local maximum at \( x = -4 \) with a peak at approximately \( f'(-4) = 5 \).
- A local minimum at \( x = 0 \).
- Another local maximum at \( x = 4 \) with a peak at approximately \( f'(4) = 3 \).
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#### Tasks and Solutions:
(i) **Identify the critical numbers of \( f \).**
- Critical numbers occur where \( f' = 0 \) or \( f' \) is undefined.
- From the graph, \( f' \) crosses the \( x \)-axis at \( x = -2 \), \( x = 0 \), and \( x = 3 \).
- **Answer:** \( x = -2, 0, 3 \)
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(ii) **Identify the open interval(s) on which \( f \) is increasing or decreasing.**
- A function \( f \) is increasing where \( f' > 0 \) and decreasing where \( f' < 0 \).
**Increasing Intervals:**
- \( f' > 0 \) on the intervals \( (-\infty, -2) \) and \( (3, \infty) \).
**Decreasing Intervals:**
- \( f' < 0 \) on the intervals \( (-2, 0) \) and \( (0, 3) \).
- **Answers:**
- Increasing: \( (-\infty, -2) \), \( (3, \in
Transcribed Image Text:### Analysis of Function Behavior
#### Critical Points and Behavior Analysis
Consider the graph of \( f'(x) \) provided. To analyze the behavior of the function \( f(x) \), we need to identify critical points, increasing and decreasing intervals, and relative extrema.
**Graph Analysis:**
- The graph provided is of \( f' \) (the first derivative of \( f \)).
- The x-axis ranges from -4 to 4, and the y-axis ranges from -4 to 4.
- The graph of \( f' \) intersects the x-axis at points where \( f'(x) = 0 \).
**(i) Identify the Critical Numbers:**
Critical numbers are values of \( x \) where \( f'(x) = 0 \) or where \( f'(x) \) does not exist.
**(ii) Identify the Open Intervals of Increase/Decrease:**
Determine the intervals on the x-axis where \( f'(x) \) is positive (indicating \( f(x) \) is increasing) and where \( f'(x) \) is negative (indicating \( f(x) \) is decreasing).
**(iii) Determine Relative Extrema:**
Evaluate where the function \( f(x) \) has relative maximum and minimum points:
- A relative maximum occurs where \( f'(x) \) changes from positive to negative.
- A relative minimum occurs where \( f'(x) \) changes from negative to positive.
#### Graph Details:
- \( f' \) crosses the x-axis, suggesting critical numbers.
- Identify intervals of positivity and negativity of \( f' \).
#### Questions:
1. **Identify the critical numbers of \( f \). (Enter your answers as a comma-separated list.)**
\[
x = \_\_\_\_\_\_
\]
2. **Identify the open interval(s) on which \( f \) is increasing or decreasing. (Enter your answers using interval notation.)**
- Increasing: \(\_\_\_\_\_\_)
- Decreasing: \(\_\_\_\_\_\_\)
3. **Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical number.**
- Relative Minimum: \( x = \_\_\_\_\_\_\)
- Relative Maximum: \( x = \_\_\_\_\_\_\)
Please refer to the provided graph to
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
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![### Analysis of Function Behavior
#### Critical Points and Behavior Analysis
Consider the graph of \( f'(x) \) provided. To analyze the behavior of the function \( f(x) \), we need to identify critical points, increasing and decreasing intervals, and relative extrema.
**Graph Analysis:**
- The graph provided is of \( f' \) (the first derivative of \( f \)).
- The x-axis ranges from -4 to 4, and the y-axis ranges from -4 to 4.
- The graph of \( f' \) intersects the x-axis at points where \( f'(x) = 0 \).
**(i) Identify the Critical Numbers:**
Critical numbers are values of \( x \) where \( f'(x) = 0 \) or where \( f'(x) \) does not exist.
**(ii) Identify the Open Intervals of Increase/Decrease:**
Determine the intervals on the x-axis where \( f'(x) \) is positive (indicating \( f(x) \) is increasing) and where \( f'(x) \) is negative (indicating \( f(x) \) is decreasing).
**(iii) Determine Relative Extrema:**
Evaluate where the function \( f(x) \) has relative maximum and minimum points:
- A relative maximum occurs where \( f'(x) \) changes from positive to negative.
- A relative minimum occurs where \( f'(x) \) changes from negative to positive.
#### Graph Details:
- \( f' \) crosses the x-axis, suggesting critical numbers.
- Identify intervals of positivity and negativity of \( f' \).
#### Questions:
1. **Identify the critical numbers of \( f \). (Enter your answers as a comma-separated list.)**
\[
x = \_\_\_\_\_\_
\]
2. **Identify the open interval(s) on which \( f \) is increasing or decreasing. (Enter your answers using interval notation.)**
- Increasing: \(\_\_\_\_\_\_)
- Decreasing: \(\_\_\_\_\_\_\)
3. **Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical number.**
- Relative Minimum: \( x = \_\_\_\_\_\_\)
- Relative Maximum: \( x = \_\_\_\_\_\_\)
Please refer to the provided graph to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57fbba33-bf9a-44b6-8e9b-4442a3982aed%2Faddde19d-d79f-4e0b-8f19-e661293fb238%2Fgdn4hn7_reoriented.jpeg&w=3840&q=75)
