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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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**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.
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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