Problem 125 - Calculus Volume 1

Question

For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. $$y=|x+1|+|x-1| \text { over }[-3,2]$$

Step-by-Step Solution

Answer

The absolute maximum is 8 at \(x = -3\).

Step 1: Understand the Function and Domain

The function given is the sum of two absolute value functions, \(|x+1| + |x-1|\), and we need to find its maxima over the interval \([-3, 2]\). Absolute value functions have distinct break points. Identify these break points, \(x = -1\) and \(x = 1\), where the function changes its linear form.

Step 2: Break Down the Function by Intervals

Divide the domain based on the absolute value terms' break points into intervals: \([-3, -1]\), \([-1, 1]\), and \([1, 2]\). Each interval will have a different expression for the function depending on the sign of each component.

Step 3: Evaluate the Function Over Each Interval

1. **Interval \([-3, -1]\):** - Here, both \(x+1\) and \(x-1\) are negative, so the function is \(f(x) = -(x+1) - (x-1) = -2x + 2\). - Evaluate at endpoints: \(f(-3) = 8\), \(f(-1) = 4\). 2. **Interval \([-1, 1]\):** - Here, \(x+1\) is positive and \(x-1\) is negative, so the function is \(f(x) = (x+1) - (x-1) = 2\). - The function is constant, so \(f(-1) = 2\) and \(f(1) = 2\).3. **Interval \([1, 2]\):** - Here, both \(x+1\) and \(x-1\) are positive, so the function is \(f(x) = x+1 + x-1 = 2x\). - Evaluate at endpoints: \(f(1) = 2\), \(f(2) = 4\).

Step 4: Determine Maximum Values

Compare the function values at all key points and intervals. From the evaluations, we found:- At \(x = -3\), \(f(x) = 8\)- At \(x = -1, f(x) = 4\)- At \(x = 1, f(x) = 2\)- At \(x = 2, f(x) = 4\)The highest value within this range is 8.

Key Concepts

Absolute Maxima

The absolute maxima refer to the highest value a function reaches over its entire domain. Unlike local maxima, which only need to be the highest value in a small neighborhood, absolute maxima mean the highest point in a given range.

In the exercise, the function \[ y = |x+1| + |x-1| \] is evaluated over the interval \([-3, 2]\).

To find the absolute maximum, you need to:

• Evaluate the function at key points: the endpoints of the interval \([-3, -1, 1, 2]\)
• Check places where the function changes form

The absolute maximum in this case occurs when \(x = -3\) and you have \(f(x) = 8\).

Understanding how to find the absolute maxima helps when analyzing real-world problems like optimizing resources or determining the highest peak in a graph.

Absolute Value Functions

Absolute value functions have a unique characteristic: They measure the "distance" of a number from zero, but never result in a negative value. Mathematically, the absolute value of a number \(x\) is denoted as \(|x|\), and this equals to \(x\) when \(x\) is positive or zero, and \(-x\) when \(x\) is negative.

In our exercise, the function is \[ y = |x+1| + |x-1| \] This represents two absolute value functions added together. The break points or changes in direction occur at \(x = -1\) and \(x = 1\).

Understanding absolute value functions is essential for simplifying expressions and solving equations that adapt based on the sign—or positivity—of the variables.

• They create V-shaped graphs
• Provide solutions that often lead to piecewise functions

Mastering this concept makes it easier to tackle complex calculus problems by breaking them down into manageable parts based on different intervals.

Piecewise Functions

Piecewise functions are essentially functions defined by multiple sub-functions, each applied to a certain interval of the main function's domain. In simpler words, they are broken into "pieces," each valid over specified intervals.

For the exercise, we broke the function \[ y=|x+1| + |x-1| \] into different intervals:

• \([-3, -1]\)
• \([-1, 1]\)
• \([1, 2]\)

Each interval has its own linear expression for evaluation.

Decomposing the function this way allows you to calculate the value of \(y\) more efficiently in each range, such as when both terms are positive or negative.
Understanding the structure of piecewise functions:

• Streamlines calculations
• Helps grasp various behaviors of a function
• Simplifies finding maxima and minima

Mastering piecewise functions opens the door to a variety of applications in mathematics and engineering.

Interval Analysis

Interval analysis involves examining a function within certain ranges or "intervals" of its domain. It's useful for functions like the absolute value function where behavior differs in separate intervals.

In our case, the function \[ y = |x+1| + |x-1| \] was split into three intervals:

• \([-3, -1]\)
• \([-1, 1]\)
• \([1, 2]\)

By analyzing each interval separately, we efficiently determine the behavior of the function, ensuring that our calculations are precise.

The benefit of interval analysis is its ability to:

• Identify points where the function changes behavior
• Provide a detailed look at the function's variations
• Simplify complex calculations by narrowing focus to smaller, manageable sections

Using this approach empowers you with the skills to tackle varied mathematical problems, unlocking new ways to optimize and solve equations in practical scenarios.