Given the graph of the function f below, determine all possible inferences that can be made about f and f' when x = 2. 8 6 5 4 3 2 -1 -10 -9 48 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 Á có -4 -5 X y -7 -8 1 2 3 Graph of f 4 5 6 7 8 10 Note: Even though we cannot technically say a function is increasing or decreasing at a point, you should indicate that either function is increasing or decreasing at x = 2 if it is doing so in a small interval surrounding x = 2. X
Given the graph of the function f below, determine all possible inferences that can be made about f and f' when x = 2. 8 6 5 4 3 2 -1 -10 -9 48 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 Á có -4 -5 X y -7 -8 1 2 3 Graph of f 4 5 6 7 8 10 Note: Even though we cannot technically say a function is increasing or decreasing at a point, you should indicate that either function is increasing or decreasing at x = 2 if it is doing so in a small interval surrounding x = 2. X
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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Given the graph of the function f below, determine all possible inferences that can be made about f and f' when x=2
Transcribed Image Text:### Analysis of the Graph of Function \( f \) at \( x = 2 \)
**Graph Overview:**
The graph provided represents the function \( f \) over the interval, with \( x \)-coordinates ranging from -10 to 10. The graph features smooth curves with peaks and troughs, indicating where the function increases, decreases, and changes direction.
**Key Observations at \( x = 2 \):**
- At \( x = 2 \), the function \( f(x) \) has a local minimum point. This suggests that the slope of the tangent (derivative) at this point, \( f'(x) \), is zero because the curve changes direction from decreasing to increasing.
- Immediately surrounding \( x = 2 \), the graph moves from decreasing (left of \( x = 2 \)) to increasing (right of \( x = 2 \)).
**Inferences on \( f \) and \( f' \):**
1. **Function \( f \) Behavior:**
- \( f \) is decreasing just before \( x = 2 \) and begins increasing right after \( x = 2 \).
2. **Derivative \( f' \) Behavior:**
- \( f'(2) = 0 \): The derivative \( f' \) at \( x = 2 \) is zero due to the local minimum.
- \( f' \) changes from negative to positive as \( x \) passes through 2, indicating a shift from decreasing to increasing.
**Note:**
Although a function cannot be said to increase or decrease at a specific point, it is important to mention the general behavior in the small interval surrounding \( x = 2 \).
This analysis captures both the behavior of the function \( f \) and its derivative \( f' \) in the vicinity around \( x = 2 \).
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