Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Identify the critical points and find the maximum value and minimum value on the given interval. $$ a(x)=|x-1| ; I=[0,3] $$

Step-by-Step Solution

Answer

Minimum is 0 at \( x=1 \); Maximum is 2 at \( x=3 \).

Step 1: Understand the Function and Interval

We are given the function \( a(x) = |x - 1| \) and the interval \( I = [0, 3] \). The task is to find the critical points of the function within this interval and to identify the maximum and minimum values.

Step 2: Break Down the Absolute Value Function

The function \( a(x) = |x - 1| \) can be broken down into two cases:- When \( x - 1 \geq 0 \) (i.e., \( x \geq 1 \)): \( a(x) = x - 1 \).- When \( x - 1 < 0 \) (i.e., \( x < 1 \)): \( a(x) = 1 - x \).

Step 3: Find Critical Points

Critical points occur where the derivative is zero or undefined. We don't need to differentiate an absolute value function for this task. Instead, observe the transition point where \( x = 1 \). Since the piecewise parts of the function, \( x - 1 \) and \( 1 - x \), are both linear, any extremum on \( I = [0, 3] \) will occur either at the endpoints of the interval or at this transition point, \( x = 1 \).

Step 4: Evaluate the Function at Critical Points and Endpoints

Evaluate \( a(x) \) at the endpoints of the interval and at the point of transition:- \( a(0) = |0 - 1| = 1 \)- \( a(1) = |1 - 1| = 0 \)- \( a(3) = |3 - 1| = 2 \)

Step 5: Determine Maximum and Minimum Values

By comparing the values: - Minimum value is \( a(1) = 0 \) - Maximum value is \( a(3) = 2 \)These give us the minimum and maximum values on the interval \( I=[0,3] \).

Key Concepts

Absolute Value Function

Understanding the absolute value function is crucial when dealing with expressions like \( a(x) = |x-1| \). An absolute value function, denoted by \( |x| \), outputs the distance of a number \( x \) from zero on the real number line, effectively removing any negative sign. In our function \( a(x) = |x - 1| \), it represents the distance of \( x \) from 1.
The behavior of an absolute value function is characterized by its ability to "flip" parts of a graph to maintain non-negative values. This feature results in a characteristic 'V' shape when graphed, centered at the point where the inside expression equals zero. For our example, \( |x - 1| \), this pivotal point is \( x = 1 \), where the function shifts from decreasing to increasing or vice versa.

Piecewise Function

A piecewise function consists of multiple sub-functions, each of which applies to a different segment of the domain. In other words, instead of having a single expression that substitutes for \( x \) across its entire range, it uses different equations depending on the value of \( x \).
With the function \( a(x) = |x-1| \), we see that it can be viewed as a piecewise function based on whether \( x \) is less than or greater than 1:

• For \( x \geq 1 \), the expression inside the absolute value is non-negative, so \( a(x) = x - 1 \).
• For \( x < 1 \), the expression inside the absolute value becomes negative, thus \( a(x) = 1 - x \).

This format simplifies the process of understanding how the function behaves over different intervals, helping us identify critical points and compute values more effectively.

Maximum and Minimum Values

Maximum and minimum values are key aspects of analyzing functions over a given interval. These values represent the highest and lowest outputs a function can produce within the specified range.
To find these values for \( a(x) = |x-1| \) over the interval \( I = [0,3] \), we first determine the function’s behavior at possible critical points and interval endpoints. In this case, the critical point stems from the absolute value transition at \( x = 1 \).
Evaluation at the endpoints \( x = 0 \) and \( x = 3 \), as well as the transition point \( x = 1 \), yields:

• Minimum value of \( a(x) = 0 \) occurs at \( x = 1 \).
• Maximum value of \( a(x) = 2 \) occurs at \( x = 3 \).

By comparing these evaluated values, we can confidently identify the minimum as occurring at \( x = 1 \) and the maximum at \( x = 3 \) within the interval.

Interval Analysis

Interval analysis involves examining how a function behaves over a specific section of its domain. This analysis helps in finding where maxima and minima occur, especially within confined ranges.
In the example \( a(x) = |x-1| \), interval analysis focuses on the domain \( I = [0, 3] \). We look at:

• The endpoints \( x = 0 \) and \( x = 3 \), where the function is evaluated for boundary values.
• The transition point \( x = 1 \), where the absolute value causes a shift between sub-functions.

The calculated values at these strategic points illustrate how interval analysis helps in identifying the overall behavior and turning points of the function within the interval \( I \). By doing so, we better understand the entire landscape of function behavior in this confined space.