Problem 79 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) is defined on a closed interval \([a, b]\), then \(f\) has an absolute maximum value.

Step-by-Step Solution

Answer

The statement is false. A counterexample is the function \(f(x) = x^2\) defined on the semi-open interval \([-1, 1)\). This function does not have an absolute maximum value, even though it is defined on a closed interval.

Step 1: Understanding the Concepts

Before we tackle the given statement, let's first understand the concepts involved. An absolute maximum value is the highest output a function can achieve within a specific domain. In our case, the domain is the closed interval [a, b]. A function defined on a closed interval is a function that has input values within that interval, including its endpoints.

Step 2: Applying the Extreme Value Theorem

Now that we understand the concepts, let's attempt to prove or disprove the statement. To do this, we can make use of the Extreme Value Theorem (EVT). The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then the function has both an absolute maximum value and an absolute minimum value on the interval. However, the given statement does not specify that the function is continuous, only that it is defined on the closed interval [a, b]. So, our objective now is to find an example of a function that is defined on a closed interval but doesn't have an absolute maximum value.

Step 3: Finding a Counterexample

Consider the function \(f(x) = x^2\) on the closed interval \([-1, 1]\). This function is continuous on the interval, and as per the EVT, it has an absolute maximum value (which is 1). Now, let's modify the function slightly by including a hole at one end: \(f(x) = x^2\) for \(x \in [-1, 1)\). This is no longer a continuous function on the closed interval [-1, 1] as it has a hole at \(x = 1\). For this modified function, we can see that no matter how close we get to \(x = 1\) from the left in this interval, we can always find a value of x closer to 1 that results in a higher output value. Thus, the modified function does not have an absolute maximum value on the interval.

Step 4: Conclusion

The given statement is false. As a counterexample, we found the function \(f(x) = x^2\) defined on the semi-open interval \([-1, 1)\). This function does not have an absolute maximum value, even though it is defined on a closed interval.

Key Concepts

Continuous Functions

Continuous functions are key in determining the behavior of a function over an interval. A function is continuous on an interval if, intuitively, you can draw it without lifting your pencil from the paper. This means there are no breaks, jumps, or holes within the interval.
Continuous functions play an important role in the Extreme Value Theorem. They ensure that maximum and minimum values do exist over closed intervals.

• Continuous functions have no breaks.
• They are predictable on given intervals.
• Crucial for applications like optimization and real-world modeling.

Understanding continuity helps in grasping why certain functions have maximum or minimum points on specific intervals.

Absolute Maximum

The absolute maximum of a function represents the highest point over a given interval. Imagine hiking on a mountain within a certain range; the absolute maximum is the peak within those bounds.
An absolute maximum might occur at:

• Endpoints of the interval.
• A critical point within the interval.

It's crucial to differentiate between absolute maximum, found over an entire interval, and local maximum, found in smaller sections.

Closed Interval

A closed interval \[a, b\] includes all numbers from \(a\) to \(b\) including the endpoints \(a\) and \(b\). It is represented with square brackets which show that the boundary points are part of the interval.
Closed intervals are particularly relevant for continuous functions:

• Guaranteed to have absolute extrema if function is continuous.
• Examine behavior at endpoints for extrema.

This makes closed intervals suitable for problems where extreme values are essential, like maximizing profits or minimizing costs.

Counterexample

A counterexample demonstrates that a general claim is false by providing a specific instance where it does not hold. In mathematical proofs, counterexamples are invaluable tools for testing hypotheses.
Considerations when finding a counterexample:

• Modify known functions by introducing discontinuities.
• Use extreme cases or endpoints to test limitations.

For example, altering the continuity of a function like \(f(x) = x^2\) by removing an endpoint can show the absence of an absolute maximum, despite being defined on a closed interval, illustrating the importance of conditions like continuity.