Problem 11 - Calculus for Biology and Medicine

Question

Sketch the graph of a function that is continuous on the open interval \((0,1)\) and has a global maximum but does not have a global minimum.

Step-by-Step Solution

Answer

The graph of \(f(x) = -\log(x)\) on \((0,1)\) illustrates a global maximum approaching 0 at \(x=1\) and no global minimum.

Step 1: Understand the Properties of the Function

The function needs to be continuous on the open interval \((0,1)\). This means that the graph should have no breaks or holes in this interval. Additionally, it must have a global maximum, meaning there should be a highest point on the interval, but no global minimum, indicating it should not have a lowest point in the graph.

Step 2: Conceptualize the Interval Endpoint Behavior

Since the function is defined only on \((0,1)\) and the open interval does not include the endpoints, the function can approach these endpoints in a manner that would allow it to avoid having a global minimum.

Step 3: Choose a Possible Function

Consider a function such as \(f(x) = -\log(x)\). This function is continuous on \((0,1)\). As \(x\) approaches 0 from the right, \(-\log(x)\) approaches infinity, avoiding a minimum. On the other hand, it achieves a maximum as \(x\) approaches 1, since \(-\log(1) = 0\), which is the highest value over this interval.

Step 4: Sketch the Function Based on Your Expression

On a graph, \(f(x) = -\log(x)\) starts from very high values near the left end of the interval \(x \to 0^+\) and decreases as it approaches the right end where \(x = 1\). The function achieves its maximum at \(x=1\), and since there is no smallest value as \(x\) approaches 0, there is no minimum.

Key Concepts

Global Maximum

A global maximum is the highest point on the graph of a function within a given interval. It is the point where the function attains its largest value. In mathematical terms, if a function \( f(x) \) has a global maximum at \( x = c \), then \( f(c) \) is greater than or equal to \( f(x) \) for all \( x \) in the considered interval.

In the context of the exercise, we're focusing on the open interval \((0,1)\). The global maximum occurs when the function reaches its peak within this interval. Specifically, using \( f(x) = -\log(x) \), the function achieves its global maximum at \( x = 1 \) because \( -\log(1) = 0 \), which is the largest value attainable in this scenario and is the point at which the downward slope of the curve starts. Even though \( x = 1 \) is not part of the open interval, approaching it gives the highest continuous result for \( f(x) \) on \((0,1)\).

This concept is crucial in calculus and graph theory and helps in understanding how a graph behaves over an interval.

Open Interval

An open interval, such as \((0,1)\), is a set of real numbers where the endpoints are not included. This means that the values of the function are considered for all \(x\) strictly between 0 and 1, but not at 0 or 1 themselves.

For the function to be continuous on this interval, there should be no breaks, jumps, or holes in this range. It ensures that for any two points in the interval, one can find a value of the function corresponding to any intermediate point without any discontinuity.

In the exercise, the choice of the function \(-\log(x)\) is significant. As \(x\) approaches 0, \(-\log(x)\) heads toward positive infinity, indicating no minimum is reached. Approaching an open endpoint like 0 without including it allows for the graph to rise infinitely, demonstrating the importance of open intervals in allowing behaviors such as not having a defined minimum value in the given context. This concept is often used to explore various properties and behaviors of functions.

Graph Sketching

Graph sketching is the process of manually drawing a rough graph of a mathematical function based on its properties, such as continuity and extrema, without necessarily plotting each point exactly.

For the specific exercise, sketching the graph of \( f(x) = -\log(x) \) involves understanding its behavior over the interval \((0,1)\).

• The function decreases as \( x \) increases, meaning it starts from high values and slopes down as \( x \) moves right.
• As \( x \to 0^+ \), \( f(x) = -\log(x) \) grows infinitely large, indicating no minimum.
• At the point \( x=1 \) (although not included in the interval), the function reaches \( f(x) = 0 \), marking where it achieves its global maximum near the endpoint.

This sketch allows us to visualize how a function behaves in an interval, showing trends and important features like maximums and minimums. It is a valuable tool in anticipating how functions perform and can help in better understanding their real-world implications or applications.

Logarithmic Functions

Logarithmic functions are functions of the form \( f(x) = \log(x) \), where \( \log(x) \) often stands for the logarithm of \( x \) with a specified base. In this exercise, \( f(x) = -\log(x) \) is used, which reverses the sign of the usual logarithmic output.

Key characteristics of logarithmic functions include:

• They are defined only for positive values of \( x \).
• They pass through the point \( (1, 0) \), where \( \log(1) = 0 \).
• As \( x \) approaches 0 from the positive side, \( \log(x) \) tends towards negative infinity, providing \(-\log(x)\) an infinite positive value.

This makes logarithmic functions particularly interesting when considering behavior over intervals like \((0,1)\), such as the open interval examined in the exercise. Logarithmic functions like \( -\log(x) \) are especially relevant in problems involving rapid increases or decreases, logarithmic scales (like pH or decibels), and in explaining phenomena in fields from finance to science. Understanding their unique properties is important for recognizing their applications.