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:**
Given the function \( f(x) = \frac{x}{x^2 + 4} \) on the interval \([-1, 4]\), find the critical points of the function.
**Explanation:**
To find the critical points, determine where the derivative of the function equals zero or is undefined. This involves calculating \( f'(x) \) and solving for \( x \).
**Step-by-step Process:**
1. Differentiate \( f(x) \) using the quotient rule \(\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\), where \( u = x \) and \( v = x^2 + 4 \).
2. Calculate the derivative \( f'(x) \).
3. Set \( f'(x) = 0 \) and solve for \( x \) within the interval \([-1, 4]\).
4. Determine if there are any points where \( f'(x) \) is undefined within this interval.
Ensure to verify the critical points by checking within the given interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ef40cf5-8815-4e5d-9050-3d49fb5b3de7%2F8eb64e24-a8c4-488a-bd00-d159d6d340bc%2F2c1psx7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Given the function \( f(x) = \frac{x}{x^2 + 4} \) on the interval \([-1, 4]\), find the critical points of the function.
**Explanation:**
To find the critical points, determine where the derivative of the function equals zero or is undefined. This involves calculating \( f'(x) \) and solving for \( x \).
**Step-by-step Process:**
1. Differentiate \( f(x) \) using the quotient rule \(\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\), where \( u = x \) and \( v = x^2 + 4 \).
2. Calculate the derivative \( f'(x) \).
3. Set \( f'(x) = 0 \) and solve for \( x \) within the interval \([-1, 4]\).
4. Determine if there are any points where \( f'(x) \) is undefined within this interval.
Ensure to verify the critical points by checking within the given interval.
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