The figure below shows the graph of f', the derivative of f, on the interval [-5, 4]. The function f is differentiable on the interval and f(-4) = 0. (a) Find f'(-2) and f"(-2). f'(-2) = f"(-2) = -5-4-3-2 X = (b) At which x-values does f have a relative extrema on the interval (-5, 0)? (Enter your answers as a comma-separated list.) 12 (c) Find all intervals on which the graph of f is concave downward. (Enter your answer using interval notation.) X = (d) Find the x-coordinate of each of the points of inflection of the graph of f. (Enter your answers as a comma-separated list.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Graph of the Derivative \( f' \) and Analysis**

The figure above illustrates the graph of \( f' \), the derivative of the function \( f \), on the interval \([-5, 4]\). It is given that the function \( f \) is differentiable on this interval and that \( f(-4) = 0 \).

**Graph Description:**
- The x-axis ranges from -5 to 4.
- The y-axis ranges from -2 to 4.
- Key points on the graph include:
  - \( f'(-2) = 0 \)
  - \( f'(1) = 0 \)

**Questions:**

(a) **Find \( f'(-2) \) and \( f''(-2) \).**
- \( f'(-2) = \) [Enter answer]
- \( f''(-2) = \) [Enter answer]

(b) **At which x-values does \( f \) have a relative extrema on the interval \((-5, 0)\)?** (Enter your answers as a comma-separated list.)
- \( x = \) [Enter answer]

(c) **Find all intervals on which the graph of \( f \) is concave downward.** (Enter your answer using interval notation.)
- [Enter answer]

(d) **Find the x-coordinate of each of the points of inflection of the graph of \( f \).** (Enter your answers as a comma-separated list.)
- \( x = \) [Enter answer]

**Graph Analysis:**
- There is a change in direction at key points where the graph intersects the x-axis. 
- Use these points to determine concavity and points of inflection.
Transcribed Image Text:**Graph of the Derivative \( f' \) and Analysis** The figure above illustrates the graph of \( f' \), the derivative of the function \( f \), on the interval \([-5, 4]\). It is given that the function \( f \) is differentiable on this interval and that \( f(-4) = 0 \). **Graph Description:** - The x-axis ranges from -5 to 4. - The y-axis ranges from -2 to 4. - Key points on the graph include: - \( f'(-2) = 0 \) - \( f'(1) = 0 \) **Questions:** (a) **Find \( f'(-2) \) and \( f''(-2) \).** - \( f'(-2) = \) [Enter answer] - \( f''(-2) = \) [Enter answer] (b) **At which x-values does \( f \) have a relative extrema on the interval \((-5, 0)\)?** (Enter your answers as a comma-separated list.) - \( x = \) [Enter answer] (c) **Find all intervals on which the graph of \( f \) is concave downward.** (Enter your answer using interval notation.) - [Enter answer] (d) **Find the x-coordinate of each of the points of inflection of the graph of \( f \).** (Enter your answers as a comma-separated list.) - \( x = \) [Enter answer] **Graph Analysis:** - There is a change in direction at key points where the graph intersects the x-axis. - Use these points to determine concavity and points of inflection.
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