Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to find the following. 7 – e!/x 7 + el/x f(x) The intervals where the function is increasing. (Select all that apply.) O (-7, 7) O (-∞, 0) (0, 0) O (-∞, ∞) O none of these The inflection points of the function. (Round your answers to two decimal places.) (х, у) %3 (smaller x-value) (х, у) %3D (larger x-value) The intervals where the function is concave up. (Select all that apply.) O (-∞, -0.48) O (-0.48, 0) O (0, 0.30) O (0.30, ∞) O (-0.48, 0.30) The intervals where the function is concave down. (Select all that apply.) (-0, -0.48) (-0.48, 0) (0, 0.30) O (0.30, ∞) O (-0.48, 0.30)

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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**Instructions for Graph Analysis with Derivatives**

To analyze the function \( f(x) = \frac{7 - e^{1/x}}{7 + e^{1/x}} \), use a computer algebra system to graph \( f \) and determine its first and second derivatives, \( f' \) and \( f'' \). 

**Tasks:**

1. **Intervals of Increase**
   - Identify where the function is increasing. Choose all applicable intervals:
     - \((-7, 7)\)
     - \((-\infty, 0)\)
     - \((0, \infty)\)
     - \((-\infty, \infty)\)
     - None of these

2. **Inflection Points**
   - Find the inflection points and provide to two decimal places:
     - \( (x, y) = (\_\_\_\_, \_\_\_\_) \) (smaller x-value)
     - \( (x, y) = (\_\_\_\_, \_\_\_\_) \) (larger x-value)

3. **Intervals of Concavity**
   - Identify where the function is concave up. Choose all applicable intervals:
     - \((-\infty, -0.48)\)
     - \((-0.48, 0)\)
     - \((0, 0.30)\)
     - \((0.30, \infty)\)
     - \((-0.48, 0.30)\)

   - Identify where the function is concave down. Choose all applicable intervals:
     - \((-\infty, -0.48)\)
     - \((-0.48, 0)\)
     - \((0, 0.30)\)
     - \((0.30, \infty)\)
     - \((-0.48, 0.30)\)

**Graphical Analysis:**

- Use the graphs of \( f' \) and \( f'' \) to determine the intervals and inflection points.
- \( f' \) helps identify intervals of increase (where \( f' > 0 \)) and decrease (where \( f' < 0 \)).
- \( f'' \) aids in finding concavity: concave up (where \( f'' > 0 \)) and concave down (where \( f'' < 0 \)). 
- Inflection points occur
Transcribed Image Text:**Instructions for Graph Analysis with Derivatives** To analyze the function \( f(x) = \frac{7 - e^{1/x}}{7 + e^{1/x}} \), use a computer algebra system to graph \( f \) and determine its first and second derivatives, \( f' \) and \( f'' \). **Tasks:** 1. **Intervals of Increase** - Identify where the function is increasing. Choose all applicable intervals: - \((-7, 7)\) - \((-\infty, 0)\) - \((0, \infty)\) - \((-\infty, \infty)\) - None of these 2. **Inflection Points** - Find the inflection points and provide to two decimal places: - \( (x, y) = (\_\_\_\_, \_\_\_\_) \) (smaller x-value) - \( (x, y) = (\_\_\_\_, \_\_\_\_) \) (larger x-value) 3. **Intervals of Concavity** - Identify where the function is concave up. Choose all applicable intervals: - \((-\infty, -0.48)\) - \((-0.48, 0)\) - \((0, 0.30)\) - \((0.30, \infty)\) - \((-0.48, 0.30)\) - Identify where the function is concave down. Choose all applicable intervals: - \((-\infty, -0.48)\) - \((-0.48, 0)\) - \((0, 0.30)\) - \((0.30, \infty)\) - \((-0.48, 0.30)\) **Graphical Analysis:** - Use the graphs of \( f' \) and \( f'' \) to determine the intervals and inflection points. - \( f' \) helps identify intervals of increase (where \( f' > 0 \)) and decrease (where \( f' < 0 \)). - \( f'' \) aids in finding concavity: concave up (where \( f'' > 0 \)) and concave down (where \( f'' < 0 \)). - Inflection points occur
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