4. Let w(x) = |x – 4| – |x – 7|– x + 12, which can also be written as w(x) = abs(x – 4) – abs(x – 7) – x + 12. Use the procedure to find the absolute maximum and minimum values on the interval [0,8] and where they occur. The absolute value function |x| abs(x) has derivative abs' (x) = 1 for x > 0, abs'(x) = -1 for x < 0, and abs' (0) is not defined. When setting w' (x) = 0, you will be adding the numbers ±1,±1, and -1. Can they combine to give you zero?
4. Let w(x) = |x – 4| – |x – 7|– x + 12, which can also be written as w(x) = abs(x – 4) – abs(x – 7) – x + 12. Use the procedure to find the absolute maximum and minimum values on the interval [0,8] and where they occur. The absolute value function |x| abs(x) has derivative abs' (x) = 1 for x > 0, abs'(x) = -1 for x < 0, and abs' (0) is not defined. When setting w' (x) = 0, you will be adding the numbers ±1,±1, and -1. Can they combine to give you zero?
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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Question 4!
![**Educational Text: Problem-Solving with Absolute Value and Polynomial Functions**
**Problem 4: Absolute Value Function Analysis**
Consider the function \( w(x) = |x - 4| - |x - 7| - x + 12 \). This can be expressed as \( w(x) = \text{abs}(x - 4) - \text{abs}(x - 7) - x + 12 \).
**Objective**:
- Find the absolute maximum and minimum values on the interval \([0, 8]\).
- Determine where these values occur.
**Guidelines**:
1. Understand that the absolute value function \(|x|\) has the derivative \(\text{abs}'(x) = 1\) for \(x > 0\) and \(\text{abs}'(x) = -1\) for \(x < 0\).
2. Note that \(\text{abs}'(0)\) is not defined.
3. When solving \(w'(x) = 0\), evaluate the sum of the numbers \(\pm 1, \pm 1\), and \(-1\). Consider if they equate to zero.
**Problem 5: Polynomial Function Analysis**
Examine the function \( m(t) = (t + 2)^2(t - 3)^4 \).
**Objective**:
- Determine the absolute maximum and minimum values on the interval \([-2, 4]\).
- Identify where these values occur.
**Guidelines**:
1. Avoid expanding the expression fully. Instead, focus on factoring the derivative.
2. Completely factor the derivative before setting to zero to simplify solving for critical points.
---
**Study Notes**:
- For absolute value functions, consider the points where the expression inside the absolute value changes sign.
- For polynomial functions, critical points are found by determining where the derivative is zero or undefined.
- Evaluate endpoints and critical points to find absolute extrema.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff41b80da-c327-4960-85a3-4b9f1b2a9a75%2Ffcd55b5e-7db0-4dfa-a8c1-56df79794146%2Faqtqifn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Text: Problem-Solving with Absolute Value and Polynomial Functions**
**Problem 4: Absolute Value Function Analysis**
Consider the function \( w(x) = |x - 4| - |x - 7| - x + 12 \). This can be expressed as \( w(x) = \text{abs}(x - 4) - \text{abs}(x - 7) - x + 12 \).
**Objective**:
- Find the absolute maximum and minimum values on the interval \([0, 8]\).
- Determine where these values occur.
**Guidelines**:
1. Understand that the absolute value function \(|x|\) has the derivative \(\text{abs}'(x) = 1\) for \(x > 0\) and \(\text{abs}'(x) = -1\) for \(x < 0\).
2. Note that \(\text{abs}'(0)\) is not defined.
3. When solving \(w'(x) = 0\), evaluate the sum of the numbers \(\pm 1, \pm 1\), and \(-1\). Consider if they equate to zero.
**Problem 5: Polynomial Function Analysis**
Examine the function \( m(t) = (t + 2)^2(t - 3)^4 \).
**Objective**:
- Determine the absolute maximum and minimum values on the interval \([-2, 4]\).
- Identify where these values occur.
**Guidelines**:
1. Avoid expanding the expression fully. Instead, focus on factoring the derivative.
2. Completely factor the derivative before setting to zero to simplify solving for critical points.
---
**Study Notes**:
- For absolute value functions, consider the points where the expression inside the absolute value changes sign.
- For polynomial functions, critical points are found by determining where the derivative is zero or undefined.
- Evaluate endpoints and critical points to find absolute extrema.
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