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:**
If \( f(3) = 5 \) and \( f'(x) \geq 2 \) for \( 3 \leq x \leq 5 \), how small can \( f(5) \) possibly be?
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**Explanation:**
This problem involves finding the minimum possible value of a function \( f(x) \) at \( x = 5 \) given its value and the lower bound of its derivative over an interval.
- **Function Value at Point:** It is given that the function \( f(x) \) satisfies \( f(3) = 5 \).
- **Derivative Constraint:** The derivative \( f'(x) \) satisfies \( f'(x) \geq 2 \) for \( 3 \leq x \leq 5 \).
From this information, the goal is to determine the smallest value \( f(5) \) can take by integrating the constraint of the derivative over the interval \([3, 5]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3241d4b2-7cea-4408-ac37-56f0a05274af%2F5fa34196-e502-45b4-a837-37ad08474af5%2Fcx9xnde_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If \( f(3) = 5 \) and \( f'(x) \geq 2 \) for \( 3 \leq x \leq 5 \), how small can \( f(5) \) possibly be?
[Input Box for Answer]
**Need Help?**
- **Read It:** [Button]
- **Watch It:** [Button]
---
**Explanation:**
This problem involves finding the minimum possible value of a function \( f(x) \) at \( x = 5 \) given its value and the lower bound of its derivative over an interval.
- **Function Value at Point:** It is given that the function \( f(x) \) satisfies \( f(3) = 5 \).
- **Derivative Constraint:** The derivative \( f'(x) \) satisfies \( f'(x) \geq 2 \) for \( 3 \leq x \leq 5 \).
From this information, the goal is to determine the smallest value \( f(5) \) can take by integrating the constraint of the derivative over the interval \([3, 5]\).
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