### Problem Statement 11. (10 points) Find the absolute extrema of the function \( f(x) = \frac{1}{6}x^2 - 10x + 200 \) on the interval \([0, 48]\). Give the extreme values and the inputs at which they occur. ### Questions - \( f(x) \) has an absolute **maximum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \). - \( f(x) \) has an absolute **minimum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \). ### Instructions 1. **Determine Critical Points:** - Find the derivative of \( f(x) \). - Solve \( f'(x) = 0 \) to find critical points. 2. **Evaluate Extreme Values:** - Calculate \( f(x) \) at the critical points and the endpoints \( x = 0 \) and \( x = 48 \). 3. **Identify Absolute Extrema:** - Compare the values from the previous step to determine the absolute maximum and minimum. 4. **Complete the Statements:** - Fill in the blanks with the correct values and \( x \)-coordinates.
### Problem Statement 11. (10 points) Find the absolute extrema of the function \( f(x) = \frac{1}{6}x^2 - 10x + 200 \) on the interval \([0, 48]\). Give the extreme values and the inputs at which they occur. ### Questions - \( f(x) \) has an absolute **maximum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \). - \( f(x) \) has an absolute **minimum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \). ### Instructions 1. **Determine Critical Points:** - Find the derivative of \( f(x) \). - Solve \( f'(x) = 0 \) to find critical points. 2. **Evaluate Extreme Values:** - Calculate \( f(x) \) at the critical points and the endpoints \( x = 0 \) and \( x = 48 \). 3. **Identify Absolute Extrema:** - Compare the values from the previous step to determine the absolute maximum and minimum. 4. **Complete the Statements:** - Fill in the blanks with the correct values and \( x \)-coordinates.
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
11. (10 points) Find the absolute extrema of the function \( f(x) = \frac{1}{6}x^2 - 10x + 200 \) on the interval \([0, 48]\). Give the extreme values and the inputs at which they occur.
### Questions
- \( f(x) \) has an absolute **maximum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \).
- \( f(x) \) has an absolute **minimum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \).
### Instructions
1. **Determine Critical Points:**
- Find the derivative of \( f(x) \).
- Solve \( f'(x) = 0 \) to find critical points.
2. **Evaluate Extreme Values:**
- Calculate \( f(x) \) at the critical points and the endpoints \( x = 0 \) and \( x = 48 \).
3. **Identify Absolute Extrema:**
- Compare the values from the previous step to determine the absolute maximum and minimum.
4. **Complete the Statements:**
- Fill in the blanks with the correct values and \( x \)-coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bdec363-0864-48e9-9cb3-5716963e6c1f%2Ff5d992c6-8d46-43b0-824d-b726aad4ae8c%2Fv49bq3f.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
11. (10 points) Find the absolute extrema of the function \( f(x) = \frac{1}{6}x^2 - 10x + 200 \) on the interval \([0, 48]\). Give the extreme values and the inputs at which they occur.
### Questions
- \( f(x) \) has an absolute **maximum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \).
- \( f(x) \) has an absolute **minimum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \).
### Instructions
1. **Determine Critical Points:**
- Find the derivative of \( f(x) \).
- Solve \( f'(x) = 0 \) to find critical points.
2. **Evaluate Extreme Values:**
- Calculate \( f(x) \) at the critical points and the endpoints \( x = 0 \) and \( x = 48 \).
3. **Identify Absolute Extrema:**
- Compare the values from the previous step to determine the absolute maximum and minimum.
4. **Complete the Statements:**
- Fill in the blanks with the correct values and \( x \)-coordinates.
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