Problem 55 - Thomas Calculus

Question

In Exercises \(51-58\) , find the critical points and domain endpoints for each function. Then find the value of the function at each of these points and identify extreme values (absolute and local). $$ y=\left\\{\begin{array}{ll}{4-2 x,} & {x \leq 1} \\ {x+1,} & {x>1}\end{array}\right. $$

Step-by-Step Solution

Answer

The function has a local and absolute minimum of 2 at \( x = 1 \). No absolute maximum exists.

Step 1: Identify the Domains

The function is a piecewise function with different rules for different domains. For \( x \leq 1 \), the function is \( y = 4 - 2x \), and for \( x > 1 \), the function is \( y = x + 1 \). Hence, the critical point is at the boundary \( x = 1 \).

Step 2: Calculate Value at the Critical Point

Check the value of the function at the critical point \( x = 1 \). For \( x \leq 1 \), the function is \( y = 4 - 2 \times 1 = 2 \). Since the function is right-continuous at \( x = 1 \), for \( x > 1 \), the function is \( y = 1 + 1 = 2 \). Therefore, the function value at \( x = 1 \) is 2 for both cases.

Step 3: Calculate Endpoint Values

Examine the limits at endpoints of the domain. The left endpoint at the limit \( x \to -\infty \) for \( 4 - 2x \) approaches \(+\infty\). The right endpoint at the limit \( x \to \infty \) for \( x + 1 \) approaches \(+\infty\) as well.

Step 4: Analysis for Extrema

For \( x \leq 1 \), the function \( 4 - 2x \) is decreasing. For \( x > 1 \), the function \( x + 1 \) is increasing. Since \( x = 1 \) connects both pieces of the function and switches behavior, the critical point \( x = 1 \) is both a local and the absolute minimum, with a function value of \( 2 \). No absolute maximum exists due to the open-ended limits on both ends.

Key Concepts

Domain Endpoints

In the context of functions, domain endpoints refer to specific points in the domain where the behavior of the function might change or where it is crucial to analyze due to potential extremes. In a piecewise function like the one in our exercise, each segment of the function has its own domain. Here, the piecewise function is defined as \( y = 4 - 2x \) for \( x \leq 1 \) and \( y = x + 1 \) for \( x > 1 \).
The domain endpoints are essential as they often highlight the transition between different behaviors in the function. For this specific problem:

• The domain endpoint occurs at \( x = 1 \), where the rule changes.
• At this point, we must check continuity and any potential extrema.

If a function doesn't have specified limits, like going towards \( -\infty \) or \( \infty \), the direction or change at these endpoints is noteworthy, as it often reflects the function's behavior beyond the typical static domain.

Piecewise Function

A piecewise function is a function composed of multiple "pieces," each defined by its own formula. The formula used depends on the specific part of the domain being considered. This type of function is particularly useful when different behaviors need to be modeled under varying conditions.
For the given exercise, the piecewise function is:

• \( y = 4 - 2x \) when \( x \leq 1 \)
• \( y = x + 1 \) when \( x > 1 \)

This indicates that for values of \( x \) less than or equal to 1, the function will follow the linear equation \( 4 - 2x \), which decreases as \( x \) increases within that range.
For values of \( x \) greater than 1, the function instead follows the equation \( x + 1 \), which increases as \( x \) increases. Understanding how each piece functions independently is vital to determine the overall behavior and any critical points.

Absolute and Local Extrema

Extrema, or extreme values, in mathematics, refer to the maximum and minimum values a function can reach. These can be categorized as absolute extrema or local extrema.
Absolute extrema are the highest or lowest points over the entire domain of a function. In our piecewise example, the critical point at \( x = 1 \) is the absolute minimum since the function value, at that point, is 2 for both piecewise segments. There is no absolute maximum because the function's endpoints extend towards \( \infty \).

• Local extremum refers to a function's high or low points relative to nearby values, but not necessarily over the entire domain.
• For the given problem, \( x = 1 \) also serves as the local minimum as both segments of the function value meet and change behavior at this point.

Identifying these extrema is crucial in analyzing the behavior of functions, as it gives insight into function trends and outcomes.

Function Behavior Analysis

Understanding a function's behavior involves examining its general trends, increasing or decreasing intervals, and how it transitions between these. This concept is pivotal, especially when dealing with piecewise functions.
In our exercise, the function can be broken down into two major parts:

• The segment \( 4 - 2x \) for \( x \leq 1 \) decreases steadily. This linear decrease suggests that the function has an inverse relationship with \( x \).
• After the critical point \( x = 1 \), the segment \( x + 1 \) for \( x > 1 \) begins to increase. This indicates a direct relationship between \( x \) and the function value.

By looking at these behaviors around \( x = 1 \), where the function pivots from decreasing to increasing, we establish \( x = 1 \) as the critical point and identify its implications. Conclusively, analyzing these shifts in function behavior under different domain conditions helps predict how the function reacts or transforms progressively.