Problem 86 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

One condition that must be satisfied before Theorem 3 (page 713 ) is applicable is that the interval on which \(f\) is defined must be a closed interval \([a, b] .\) Define a function \(f\) on the open interval \((-1,1)\) by \(f(x)=x\). Show that \(f\) does not attain an absolute maximum or an absolute minimum on the interval \((-1,1)\). Hint: What happens to \(f(x)\) if \(x\) is close to but not equal to \(x=\) \(-1 ?\) If \(x\) is close to but not equal to \(x=1 ?\)

Step-by-Step Solution

Answer

The function \(f(x) = x\) defined on the open interval \((-1,1)\) does not attain an absolute maximum or minimum. As \(x\) approaches -1 or 1, the function gets arbitrarily close to these values, but never reaches them due to the open nature of the interval. Thus, \(f(x)\) does not meet the conditions necessary to apply Theorem 3 from page 713.

Step 1: Understanding the Context

We need to understand that an absolute maximum or minimum of a function on an interval is a value that the function attains on that interval that is greater than or equal to (for maximum) or less than or equal to (for minimum) all other values of the function on that interval.

Step 2: Exploring the Interval

Consider the open interval \((-1,1)\) on which the function \(f(x) = x\) is defined. Note that \(f(x)\) can take any real value between -1 and 1 (but not inclusive), which corresponds with the definition of an open interval.

Step 3: Evaluating the Endpoints

For \(x\) close to but not equal to -1, \(f(x) = x\) will likewise be close to, but greater than -1. Similarly, for \(x\) close to but not equal to 1, \(f(x) = x\) will be close to, but less than 1. In neither case does \(f(x)\) actually reach -1 or 1, since -1 and 1 do not lie within the open interval \((-1,1)\).

Step 4: Conclusion

Hence, it can be concluded that \(f(x) = x\) defined on the open interval \((-1,1)\) does not attain an absolute maximum or minimum. It approaches but never reaches -1 (for an absolute minimum) and 1 (for an absolute maximum) since the interval is open and does not include the endpoints -1 and 1. Therefore, \(f(x)\) does not meet the condition necessary to apply Theorem 3 from page 713.

Key Concepts

Functions

A function is a special relationship where each input has a single output. It is often represented as a rule that maps an input value ( x ) to an output value ( f(x) ). This concept is fundamental in calculus as it helps us describe how one quantity changes in relation to another.

For example, the function described in this exercise is simple: f(x) = x . Here, the input ( x ) directly corresponds to the output ( f(x) ). Every point on this function is a line, making it a linear function.

Functions can take different forms and involve various operations, such as addition, subtraction, multiplication, and division.

• Linear functions, like f(x) = x , have a constant rate of change.
• Quadratic functions, like f(x) = x^2 , have a variable rate of change.
• Other complex functions can be created by combining or transforming simpler functions.

This mapping characteristic of functions is crucial to understanding many aspects of calculus.

Intervals

In mathematics, an interval is a range of numbers bounded by two endpoints. There are several types of intervals, each defined by whether these endpoints are included or not.

• A closed interval [a, b] includes both endpoints, meaning it contains all the numbers from a to b .
• An open interval (a, b) excludes both endpoints, consisting only of the values between a and b .
• Half-open intervals include only one endpoint, such as (a, b] or [a, b) .

In the given exercise, we are dealing with an open interval (-1, 1) . This means that the numbers -1 and 1 are not part of the interval, even though the function f(x) = x approaches these values. Understanding the type of interval is critical because it affects whether the function can reach specific extrema, a key point in this problem.

Absolute Extrema

Absolute extrema are the highest or lowest points that a function can reach within a given interval. These points are called the absolute maximum and absolute minimum, respectively. In a closed interval, the function must both attain these values and have endpoints.

For a function f(x) defined on an open interval like (-1, 1) , it might not have absolute extrema because the function does not achieve the boundary values. In the exercise, f(x) = x is evaluated on the open interval (-1, 1) . While f(x) gets closer to -1 (absolute minimum) and 1 (absolute maximum), it never actually attains these values.

• Absolute maximum is the largest value over the entire domain.
• Absolute minimum is the smallest value over the entire domain.

Thus, due to the open nature of the interval, the function does not reach its extrema, which is why f(x) = x has no absolute maximum or minimum in (-1, 1) .

Open Interval

An open interval (a, b) is a type of interval where the endpoints a and b are not included in the set of possible values. This concept is significant in calculus when considering the domain of a function over which we want to find extrema.

In the context of our exercise, the function f(x) = x is defined over the open interval (-1, 1) . Meaning as x approaches the boundaries of the interval, it never actually reaches -1 or 1.

This characteristic of open intervals implies:

• The function f(x) will approach but never equal the boundary values.
• Many common theorems regarding continuity and extrema may not apply, as they typically require closed intervals, options to include endpoints.

Understanding open intervals is crucial when determining if a function can achieve maximum or minimum values within a specified domain, as seen in this example where the lack of closed endpoints prevents achieving true extrema.