The graph of f'(x) is shown. Find i) The intervals where f is increasing and decreasing. ii) What values of x does f have a local minimum or maximum iii) A potential graph of f(x) in the interval -2 ≤ x ≤ 4 Use interval notation where necessary
The graph of f'(x) is shown. Find i) The intervals where f is increasing and decreasing. ii) What values of x does f have a local minimum or maximum iii) A potential graph of f(x) in the interval -2 ≤ x ≤ 4 Use interval notation where necessary
The graph of f'(x) is shown. Find i) The intervals where f is increasing and decreasing. ii) What values of x does f have a local minimum or maximum iii) A potential graph of f(x) in the interval -2 ≤ x ≤ 4 Use interval notation where necessary
i) The intervals where f is increasing and decreasing.
ii) What values of x does f have a local minimum or maximum
iii) A potential graph of f(x) in the interval -2 ≤ x ≤ 4
Use interval notation where necessary
Transcribed Image Text:The graph depicted in the image is of the derivative \( f'(x) \) of function \( f \).
### Graph Description:
- **Axes**: The graph is plotted on a coordinate plane with the horizontal axis labeled as \( x \) and the vertical axis labeled as \( y \).
- **Curve**: The curve represents \( y = f'(x) \).
- **Behavior**:
- The graph crosses the x-axis three times, indicating the points where \( f'(x) = 0 \). These points are approximately \( x = 0 \), \( x = 2 \), and \( x = 5 \).
- Between these points, the graph alternates above and below the x-axis, indicating intervals where the function \( f(x) \) is increasing (above the x-axis) or decreasing (below the x-axis).
- **Critical Points**:
- The peaks (local maxima) and troughs (local minima) of \( f'(x) \) represent points of greatest change in \( f(x) \).
- There is a local maximum around \( x = -0.5 \) and another local maximum near \( x = 4.5 \).
- A local minimum is located near \( x = 1.5 \) and another one slightly beyond \( x = 3.5 \).
- **Domain and Range**:
- The graph is shown in the domain \( -1 \leq x \leq 7 \).
- The range of \( f'(x) \) appears to be between roughly -2 and 2.
This graph provides critical insights into the behavior of the original function \( f(x) \), such as intervals of increase and decrease, and the concavity of \( f(x) \) based on the behavior of its derivative \( f'(x) \).
Formula Formula A function f ( x ) is also said to have attained a local minimum 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 a case f ( a ) is called the local minimum value of f ( x ) at x = a .
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