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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Question
![**Problem Statement:**
**How many critical numbers does \( f \) have?**
The function is defined as:
\[ f'(x) = \frac{100 \cos^2(x)}{10 + x^2} - 1 \]
**Explanation:**
To find the critical numbers of the function \( f \), determine the values of \( x \) for which \( f'(x) = 0 \) or \( f'(x) \) is undefined. Here, \( f'(x) \) is a rational function involving the cosine square function, a constant, and a quadratic term in the denominator.
**Steps:**
1. **Set \( f'(x) = 0 \) and solve for \( x \):**
\[ \frac{100 \cos^2(x)}{10 + x^2} - 1 = 0 \]
2. **Solve the equation to find \( x \).**
3. **Check for any points where \( f'(x) \) is undefined, considering the function is a rational expression.**
No graphs or diagrams are present in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F209171d0-af6c-4a82-ab43-10f07d999625%2F3e237fff-0db4-412e-aa80-f3c4723c8a37%2Foksl7ov_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
**How many critical numbers does \( f \) have?**
The function is defined as:
\[ f'(x) = \frac{100 \cos^2(x)}{10 + x^2} - 1 \]
**Explanation:**
To find the critical numbers of the function \( f \), determine the values of \( x \) for which \( f'(x) = 0 \) or \( f'(x) \) is undefined. Here, \( f'(x) \) is a rational function involving the cosine square function, a constant, and a quadratic term in the denominator.
**Steps:**
1. **Set \( f'(x) = 0 \) and solve for \( x \):**
\[ \frac{100 \cos^2(x)}{10 + x^2} - 1 = 0 \]
2. **Solve the equation to find \( x \).**
3. **Check for any points where \( f'(x) \) is undefined, considering the function is a rational expression.**
No graphs or diagrams are present in the image.
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