Problem 71 - Thomas Calculus

Question

Graph the functions in Exercises \(71-74 .\) Then find the extreme values of the function on the interval and say where they occur. $$f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5$$

Step-by-Step Solution

Answer

The minimum value is -9 at x = -5, and the maximum is 11 at x = 5.

Step 1: Understanding Absolute Value Functions

The function given is \( f(x) = |x-2| + |x+3| \). Absolute value functions are piecewise functions, which means this function can change its expression depending on the value of \( x \). Specifically, the points \( x = 2 \) and \( x = -3 \) are critical points where the expression for \( f(x) \) may change.

Step 2: Defining Piecewise Regions

Given the critical points from the absolute values, the function \( f(x) \) can be split into three regions based on \( x = -3 \) and \( x = 2 \):1. \( x < -3 \)2. \( -3 \leq x < 2 \)3. \( x \geq 2 \)

Step 3: Evaluating Each Region

For each region, evaluate \( f(x) \):1. **Region 1:** \( x < -3 \) leads to both terms flipping sign: \( f(x) = -(x-2) - (x+3) = -2x - 1 \).2. **Region 2:** \( -3 \leq x < 2 \) leads to the second term flipping sign: \( f(x) = x-2 - (x+3) = -5 \).3. **Region 3:** \( x \geq 2 \) keeps both terms positive: \( f(x) = (x-2) + (x+3) = 2x + 1 \).

Step 4: Graphing the Piecewise Function

Using the expressions identified, the function can be graphed:- For \( x < -3 \), the graph is a line with slope -2 and intersects the y-axis at -1.- For \( -3 \leq x < 2 \), the graph is constant at \( f(x) = -5 \).- For \( x \geq 2 \), the graph is a line with slope 2 and intersects the y-axis at 1.Directly draw these pieces on the defined interval \([-5, 5]\).

Step 5: Identifying Extreme Values

The extreme values can occur at the endpoints or at points where the derivative changes, which include \( x = -3 \), and \( x = 2 \). Compute the function values at these points:- At \( x = -5 \): \( f(-5) = -9 \).- At \( x = -3 \): \( f(-3) = -5 \).- At \( x = 2 \): \( f(2) = 5 \).- At \( x = 5 \): \( f(5) = 11 \).The minimum value is \(-9\) at \( x = -5 \), and the maximum value is \( 11 \) at \( x = 5 \).

Key Concepts

Piecewise Functions

Understanding piecewise functions is crucial when dealing with absolute value functions. A piecewise function is a function that is defined by different expressions over different intervals of the domain. For the function \( f(x) = |x-2| + |x+3| \), it can take on different forms based on the value of \( x \). This means that instead of having one overarching formula, you need different formulas for different sections of the number line.

In our example, the critical points are \( x = -3 \) and \( x = 2 \), where the nature of the absolute values changes. From this, the function can be split into several distinct intervals:

• For \( x < -3 \), both expressions inside the absolute values are negative, so \( f(x) = -(x-2) - (x+3) = -2x - 1 \).
• For \( -3 \leq x < 2 \), the expression \( x+3 \) is positive but \( x-2 \) is negative, making \( f(x) = (x-2)-(x+3) = -5 \).
• For \( x \geq 2 \), both expressions are positive, so \( f(x) = (x-2)+(x+3) = 2x + 1 \).

Breaking the function into these intervals allows us to precisely define the behavior of \( f(x) \) over each range, which is key to understanding its overall structure.

Extreme Values

Extreme values of a function refer to its maximum and minimum points within a specified interval. These are essential when analyzing any function as they help determine the range of function outcomes. In our function \( f(x) = |x-2| + |x+3| \), we are asked to find extreme values over the interval \([-5, 5]\).

To determine these values, we first look at the endpoints and the points where the piecewise function changes, which are often critical points. For our function, these points are:

• \( x = -5 \): Evaluate to find \( f(-5) = -9 \).
• \( x = -3 \): At this boundary between piecewise intervals, \( f(-3) = -5 \).
• \( x = 2 \): Another critical point, \( f(2) = 5 \).
• \( x = 5 \): Evaluate to find \( f(5) = 11 \).

Looking at these values, the maximum is observed at \( x = 5 \) with \( f(x) = 11 \), while the minimum occurs at \( x = -5 \) with \( f(x) = -9 \). This analytical approach highlights where \( f(x) \) reaches its highest and lowest values in the given domain.

Graphing Functions

Graphing functions, especially piecewise functions like \( f(x) = |x-2| + |x+3| \), is a powerful visual method to understand their behavior over an interval. By plotting the function, one can directly observe the changes in direction and magnitude, which help in identifying patterns and critical values.

To graph our specific function, we first plot each piece separately based on the defined intervals:

• For \( x < -3 \), graph the line \( f(x) = -2x - 1 \). This line has a negative slope of -2, indicating it decreases as \( x \) decreases, starting from the y-intercept of -1.
• For \( -3 \leq x < 2 \), plot \( f(x) = -5 \), which is a constant line across this interval, showing \( f(x) \) does not change within this range.
• For \( x \geq 2 \), graph \( f(x) = 2x + 1 \), a line with a positive slope of 2, illustrating an increase as \( x \) increases from the y-intercept of 1.

By joining these pieces according to their intervals on the domain \([-5, 5]\), the entire graph can be constructed, offering a complete picture of how the function behaves across its range. Additionally, observing the graph makes it easier to spot the points of extreme values visually, corroborating our analytical findings.