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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![### Critical Numbers of a Function
**Problem:**
Given the derivative of a function \( f \), determine the number of critical numbers the function has.
**Derivative Formula:**
\[ f'(x) = 6e^{-0.1|x|} \sin(x) - 1 \]
**Task:**
Find the number of critical numbers.
**Explanation:**
Critical numbers occur where the derivative \( f'(x) = 0 \) or where \( f'(x) \) is undefined. The expression involves an exponential decay \( 6e^{-0.1|x|} \), coupled with the sine function \(\sin(x)\), subtracted by 1.
### Steps to Find Critical Numbers:
1. **Set \( f'(x) = 0 \):**
Solve the equation:
\[ 6e^{-0.1|x|} \sin(x) - 1 = 0 \]
2. **Simplify:**
\[ 6e^{-0.1|x|} \sin(x) = 1 \]
3. **Solve for \(|x|\) and \(\sin(x)\):**
\[ e^{-0.1|x|} = \frac{1}{6 \sin(x)} \]
4. **Consider domain and behavior of \(\sin(x)\):**
Analyze where the equation holds true considering \( \sin(x) \) values and the exponential function domain.
### Conclusion:
Calculate specific values or intervals for \( x \) where this equation is satisfied to determine the critical numbers.
*Note: Further solving may require graphical or numerical methods beyond algebraic manipulation.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b5f6b20-2a90-4bf5-8874-1eafd41b02f1%2Ff94f1533-b800-4190-a9b4-8bce4338a99c%2Fbjbufnc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Critical Numbers of a Function
**Problem:**
Given the derivative of a function \( f \), determine the number of critical numbers the function has.
**Derivative Formula:**
\[ f'(x) = 6e^{-0.1|x|} \sin(x) - 1 \]
**Task:**
Find the number of critical numbers.
**Explanation:**
Critical numbers occur where the derivative \( f'(x) = 0 \) or where \( f'(x) \) is undefined. The expression involves an exponential decay \( 6e^{-0.1|x|} \), coupled with the sine function \(\sin(x)\), subtracted by 1.
### Steps to Find Critical Numbers:
1. **Set \( f'(x) = 0 \):**
Solve the equation:
\[ 6e^{-0.1|x|} \sin(x) - 1 = 0 \]
2. **Simplify:**
\[ 6e^{-0.1|x|} \sin(x) = 1 \]
3. **Solve for \(|x|\) and \(\sin(x)\):**
\[ e^{-0.1|x|} = \frac{1}{6 \sin(x)} \]
4. **Consider domain and behavior of \(\sin(x)\):**
Analyze where the equation holds true considering \( \sin(x) \) values and the exponential function domain.
### Conclusion:
Calculate specific values or intervals for \( x \) where this equation is satisfied to determine the critical numbers.
*Note: Further solving may require graphical or numerical methods beyond algebraic manipulation.*
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