The given function is \( f(x) = x + 4x^{-1} \). **Task:** Determine the intervals where \( f \) is increasing and decreasing. Place your answers in the appropriate blanks below. **Solution Steps:** To find where the function is increasing or decreasing, one must compute the derivative \( f'(x) \) and analyze its sign. 1. **Compute the derivative:** The derivative of \( f(x) = x + 4x^{-1} \) is \( f'(x) = 1 - \frac{4}{x^2} \). 2. **Set the derivative equal to zero to find critical points:** Solve \( 1 - \frac{4}{x^2} = 0 \) for \( x \). This gives \( x^2 = 4 \), so \( x = \pm 2 \). 3. **Test intervals determined by critical points:** The critical points split the number line into intervals: \( (-\infty, -2) \), \( (-2, 2) \), and \( (2, \infty) \). 4. **Determine the sign of \( f'(x) \) in each interval:** - For \( x \in (-\infty, -2) \), pick \( x = -3 \): \( f'(-3) = 1 - \frac{4}{9} = \frac{5}{9} > 0 \). - For \( x \in (-2, 2) \), pick \( x = 0 \): \( f'(0) \) is undefined (check for behavior of \( f(x) \)). - For \( x \in (2, \infty) \), pick \( x = 3 \): \( f'(3) = 1 - \frac{4}{9} = \frac{5}{9} > 0 \). 5. **Conclusion:** - The function \( f(x) \) is **increasing** on the intervals \( (-\infty, -2) \cup (2, \infty) \). - The function \( f(x) \) is **decreasing** on the interval \( (-2, 2) \), but note special attention is needed at \( x = 0 \) as it is undefined
The given function is \( f(x) = x + 4x^{-1} \). **Task:** Determine the intervals where \( f \) is increasing and decreasing. Place your answers in the appropriate blanks below. **Solution Steps:** To find where the function is increasing or decreasing, one must compute the derivative \( f'(x) \) and analyze its sign. 1. **Compute the derivative:** The derivative of \( f(x) = x + 4x^{-1} \) is \( f'(x) = 1 - \frac{4}{x^2} \). 2. **Set the derivative equal to zero to find critical points:** Solve \( 1 - \frac{4}{x^2} = 0 \) for \( x \). This gives \( x^2 = 4 \), so \( x = \pm 2 \). 3. **Test intervals determined by critical points:** The critical points split the number line into intervals: \( (-\infty, -2) \), \( (-2, 2) \), and \( (2, \infty) \). 4. **Determine the sign of \( f'(x) \) in each interval:** - For \( x \in (-\infty, -2) \), pick \( x = -3 \): \( f'(-3) = 1 - \frac{4}{9} = \frac{5}{9} > 0 \). - For \( x \in (-2, 2) \), pick \( x = 0 \): \( f'(0) \) is undefined (check for behavior of \( f(x) \)). - For \( x \in (2, \infty) \), pick \( x = 3 \): \( f'(3) = 1 - \frac{4}{9} = \frac{5}{9} > 0 \). 5. **Conclusion:** - The function \( f(x) \) is **increasing** on the intervals \( (-\infty, -2) \cup (2, \infty) \). - The function \( f(x) \) is **decreasing** on the interval \( (-2, 2) \), but note special attention is needed at \( x = 0 \) as it is undefined
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:The given function is \( f(x) = x + 4x^{-1} \).
**Task:**
Determine the intervals where \( f \) is increasing and decreasing. Place your answers in the appropriate blanks below.
**Solution Steps:**
To find where the function is increasing or decreasing, one must compute the derivative \( f'(x) \) and analyze its sign.
1. **Compute the derivative:**
The derivative of \( f(x) = x + 4x^{-1} \) is \( f'(x) = 1 - \frac{4}{x^2} \).
2. **Set the derivative equal to zero to find critical points:**
Solve \( 1 - \frac{4}{x^2} = 0 \) for \( x \).
This gives \( x^2 = 4 \), so \( x = \pm 2 \).
3. **Test intervals determined by critical points:**
The critical points split the number line into intervals: \( (-\infty, -2) \), \( (-2, 2) \), and \( (2, \infty) \).
4. **Determine the sign of \( f'(x) \) in each interval:**
- For \( x \in (-\infty, -2) \), pick \( x = -3 \): \( f'(-3) = 1 - \frac{4}{9} = \frac{5}{9} > 0 \).
- For \( x \in (-2, 2) \), pick \( x = 0 \): \( f'(0) \) is undefined (check for behavior of \( f(x) \)).
- For \( x \in (2, \infty) \), pick \( x = 3 \): \( f'(3) = 1 - \frac{4}{9} = \frac{5}{9} > 0 \).
5. **Conclusion:**
- The function \( f(x) \) is **increasing** on the intervals \( (-\infty, -2) \cup (2, \infty) \).
- The function \( f(x) \) is **decreasing** on the interval \( (-2, 2) \), but note special attention is needed at \( x = 0 \) as it is undefined
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