Problem 31 - Calculus for Biology and Medicine

Question

Graph $$ f(x)=|1-| x||, \quad-1 \leq x \leq 2 $$ and determine all local and global extrema on \([-1,2]\).

Step-by-Step Solution

Answer

Global max is 1 at \(x = 0\) and \(x = 2\); global min is 0 at \(x = -1\) and \(x = 1\). No other local extrema.

Step 1: Understand the Function

The function given is \( f(x) = |1 - |x|| \). This is a composition of absolute value functions. We will analyze how this function behaves by breaking it into different expressions based on the value of \( x \).

Step 2: Break Down the Absolute Values

The innermost absolute value \( |x| \) splits the domain into two cases: \( x \geq 0 \) and \( x < 0 \). - If \( x \geq 0 \), then \( |x| = x \), so the function simplifies to \( |1 - x| \).- If \( x < 0 \), then \( |x| = -x \), so the function becomes \( |1 + x| \).

Step 3: Define the Cases and Simplify

Now defining the outer absolute value for each case:1. **Case 1:** \( x \geq 0 \). - The function is \( |1 - x| \): - If \( x \leq 1 \), then \( |1 - x| = 1 - x \). - If \( x > 1 \), then \( |1 - x| = x - 1 \).2. **Case 2:** \( x < 0 \). - The function is \( |1 + x| \): - If \( x \geq -1 \), then \( |1 + x| = 1 + x \). - If \( x < -1 \), the function is undefined as \( x \) is outside the given interval.

Step 4: Graph the Function

We need to graph the piecewise function derived from the breakdown:- **For \( -1 \leq x < 0 \):** The function follows \( 1 + x \), a line with a slope of 1 starting at (0,1).- **For \( 0 \leq x \leq 1 \):** The function follows \( 1 - x \), a line with a slope of -1 starting at (0,1).- **For \( 1 < x \leq 2 \):** The function follows \( x - 1 \), a line with a slope of 1 starting at (1,0).

Step 5: Determine Extrema

On the graph, identify turning points and endpoints:- The function is continuous from \( x = -1 \) to \( x = 2 \).- **Endpoints:** - At \( x = -1 \), \( f(-1) = 0 \). - At \( x = 2 \), \( f(2) = 1 \).- **Local Minimum at \( x = 1 \) and \( x = -1\):** - \( f(1) = 0 \) and \( f(-1) = 0 \).- **Global Maximum**: The highest value within the interval is \( f(0) = 1 \).

Step 6: Finalize the Analysis of Extrema

Summarize the extrema as follows:- Global Maximum: \( f(x) = 1 \) at \( x = 0 \) and \( x = 2 \).- Global Minimum: \( f(x) = 0 \) at \( x = -1 \) and \( x = 1 \).- No other local extrema in the interval as the function is piecewise linear.

Key Concepts

Absolute Value Function

An absolute value function is a special type of function that takes a real number and turns it into its non-negative counterpart. It is represented as \(|x|\), which means "the distance of \(x\) from zero on the number line without considering direction". For any real number \(x\), the absolute value function is defined as follows:

• If \(x \geq 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

In the given problem, \(f(x) = |1 - |x||\), we deal with nested absolute values. First, we consider \(|x|\), which splits into two scenarios depending on the sign of \(x\). Next, the expression \(|1 - |x||\) itself becomes piecewise due to the absolute values. Understanding this layering is crucial to graphing the function and finding any extrema later in the exercise.

Local and Global Extrema

In calculus, extrema refer to the minimum and maximum values of a function. These can be local or global:

• **Local extrema** are the highest or lowest points in a small neighborhood around a point.
• **Global extrema** are the highest or lowest points in the entire domain of the function.

For the function \(f(x) = |1 - |x||\) over the interval \([-1, 2]\), we found:

• **Local Minimum**: Points where the graph's direction changes, found at \(x = -1\) and \(x = 1\) with \(f(-1) = f(1) = 0\).
• **Global Maximum**: The largest value of \(f(x)\) within the interval, occurring at \(x = 0\) and \(x = 2\), where \(f(0) = f(2) = 1\).

By identifying where both local and global extrema are, we gain insights into how the function behaves over its domain.

Piecewise Functions

A piecewise function is constructed from multiple sub-functions, each applying to a specific part of the domain. Each sub-function has its domain in which the function behaves differently. In our problem, after evaluating the nested absolute values, we define our function in parts:

• For \(-1 \leq x < 0\): \(f(x) = 1 + x\).
• For \(0 \leq x \leq 1\): \(f(x) = 1 - x\).
• For \(1 < x \leq 2\): \(f(x) = x - 1\).

These sub-functions together form the complete description of \(f(x)\) within the specified interval \([-1, 2]\). Understanding each piece helps in graphing the overall function correctly and accurately finding the extremum points.

Graphing Functions

Graphing a function involves plotting its behavior over a specified range. For our function \(f(x) = |1 - |x||\), graphing helps visualize its piecewise nature. We plot three segments that represent each sub-function over the given intervals:

• **From \(-1\) to 0**: The line \(1 + x\) starts at \(-1\) and ends at \(0, 1\).
• **From 0 to 1**: The line \(1 - x\) starts from \(0, 1\) and ends at \(1, 0\).
• **From 1 to 2**: The line \(x - 1\) begins from \(1, 0\) and ends at \(2, 1\).

Graphing reveals where the function changes its direction, which corresponds to its local extrema. It also confirms the global maximum and minimum by visually contrasting peaks and valleys of the plotted function.