If f(3) 5 and f '(x) 2 2 for 3 < x < 5, how small can f(5) possibly be? Need Help? Read It Watch It

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 \([3, 5]\).
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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