Find the absolute maximum and minimum values of the following function over the indicated interval, and indicate the x-values at which they occum f(x) = x³ + 1⁄2x² -12x+8; [-6,6] (...) The absolute maximum value is at x = (Use a comma to separate answers as needed. Round to two decimal places as needed.)
Find the absolute maximum and minimum values of the following function over the indicated interval, and indicate the x-values at which they occum f(x) = x³ + 1⁄2x² -12x+8; [-6,6] (...) The absolute maximum value is at x = (Use a comma to separate answers as needed. Round to two decimal places as needed.)
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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![**Title: Finding Absolute Maximum and Minimum Values of a Function**
**Objective:**
To determine the absolute maximum and minimum values of a given function over a specified interval and identify the x-values at which these extrema occur.
---
**Problem Statement:**
Find the absolute maximum and minimum values of the following function over the indicated interval, and indicate the x-values at which they occur.
\[ f(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 12x + 8; \quad [ -6, 6] \]
---
**Instructions:**
1. **Evaluate the function at critical points and endpoints.**
- Identify the critical points of the function within the interval [ -6, 6 ] by finding the first derivative \( f'(x) \) and setting it to zero.
- Solve \( f'(x) = 0 \) to find the critical points.
- Evaluate \( f(x) \) at these critical points and at the endpoints \( x = -6 \) and \( x = 6 \).
2. **Identify the Absolute Maximum and Minimum Values:**
- Compare the values obtained to determine which is the absolute maximum and which is the absolute minimum.
3. **Record Your Findings:**
- The absolute maximum value is __ at \( x = \)__.
- The absolute minimum value is __ at \( x = \)__.
---
**Example Calculation:**
After following the steps:
- Suppose \( f'(x) = 0 \) gives critical points \( x_1, x_2, \) etc.
- Evaluate \( f(x) \) at \( x_1, x_2 \), and at the endpoints \( x = -6 \) and \( x = 6 \).
**Result:**
The absolute maximum value is __ at \( x = \) __.
The absolute minimum value is __ at \( x = \) __.
*Note: Use a comma to separate answers as needed. Round to two decimal places as needed.*
---
**Conclusion:**
Finding the absolute maximum and minimum values of a function over a given interval involves evaluating the function at critical points and endpoints, then comparing these values to determine the highest and lowest values achieved by the function over that interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46a4dff8-7f9e-4127-9d1e-ef0349b76ae2%2Fa523bbbf-a1a5-45ce-8a5f-630b178fe1e0%2Fnr77b3d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding Absolute Maximum and Minimum Values of a Function**
**Objective:**
To determine the absolute maximum and minimum values of a given function over a specified interval and identify the x-values at which these extrema occur.
---
**Problem Statement:**
Find the absolute maximum and minimum values of the following function over the indicated interval, and indicate the x-values at which they occur.
\[ f(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 12x + 8; \quad [ -6, 6] \]
---
**Instructions:**
1. **Evaluate the function at critical points and endpoints.**
- Identify the critical points of the function within the interval [ -6, 6 ] by finding the first derivative \( f'(x) \) and setting it to zero.
- Solve \( f'(x) = 0 \) to find the critical points.
- Evaluate \( f(x) \) at these critical points and at the endpoints \( x = -6 \) and \( x = 6 \).
2. **Identify the Absolute Maximum and Minimum Values:**
- Compare the values obtained to determine which is the absolute maximum and which is the absolute minimum.
3. **Record Your Findings:**
- The absolute maximum value is __ at \( x = \)__.
- The absolute minimum value is __ at \( x = \)__.
---
**Example Calculation:**
After following the steps:
- Suppose \( f'(x) = 0 \) gives critical points \( x_1, x_2, \) etc.
- Evaluate \( f(x) \) at \( x_1, x_2 \), and at the endpoints \( x = -6 \) and \( x = 6 \).
**Result:**
The absolute maximum value is __ at \( x = \) __.
The absolute minimum value is __ at \( x = \) __.
*Note: Use a comma to separate answers as needed. Round to two decimal places as needed.*
---
**Conclusion:**
Finding the absolute maximum and minimum values of a function over a given interval involves evaluating the function at critical points and endpoints, then comparing these values to determine the highest and lowest values achieved by the function over that interval.
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