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.

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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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