Problem 57 - Calculus Early Transcendentals

Question

Suppose that \(f\) is continuous and positive-valued everywhere and that the \(x\) -axis is an asymptote for the graph of \(f,\) both as \(x \rightarrow-\infty\) and as \(x \rightarrow+\infty .\) Explain why \(f\) cannot have an absolute minimum but may have a relative minimum.

Step-by-Step Solution

Answer

An absolute minimum does not exist due to asymptotic behavior, but a relative minimum might occur within a local interval.

Step 1: Understanding Absolute Minimum

An absolute minimum of a function is the lowest point on the entire graph of that function. Since \( f \) has the \( x \)-axis as a horizontal asymptote as \( x \rightarrow -\infty \) and \( x \rightarrow +\infty \), the function values approach but never reach zero. Hence \( f(x) \) cannot achieve a global lowest value as it approaches the \( x \)-axis infinitely.

Step 2: Analyzing Positive-Valued Condition

Given that \( f \) is positive-valued everywhere, it means \( f(x) > 0 \) for all \( x \). This ensures that no part of the graph dips below or touches the \( x \)-axis, reinforcing the absence of a point at which \( f \) is minimized globally.

Step 3: Understanding Relative Minimum

A relative minimum is a point where the function value is the lowest within a neighborhood around that point. Since \( f \) is continuous, changes in the graph allow for local variations. This means there can be points where \( f \) reaches a low value relative to nearby values, even if not the lowest overall.

Step 4: Example of Relative Minimum

Suppose \( f(x) = \frac{1}{x^2+1} \). Notice that \( f(x) \) is continuous, positive, and has the \( x \)-axis as a horizontal asymptote as \( x \rightarrow \pm \infty \). At \( x = 0 \), the function achieves its maximum point, decreasing as \( x \) moves away from zero, demonstrating asymptotic behavior and providing context for a relative minimum at points near \( x = 0 \).

Key Concepts

Continuous Function

A continuous function is a type of function that has no breaks, jumps, or gaps in its graph. Simply put, if a function is continuous, you can draw it on a piece of paper without lifting your pencil. This quality is essential because it ensures that every point on the graph is defined and smoothly connected. For example, for the function \(f(x)\) that we discussed, being continuous means that as you move from left to right along the \(x\)-axis, the function has no abrupt changes.

• There are no missing points on the graph.
• Every input corresponds to exactly one output, with no interruptions.
• The graph can smoothly connect across the entire range of \(x\) values.

In our specific exercise, the function \(f(x)\) is continuous and positive everywhere, meaning it stays above the \(x\)-axis throughout its domain. This continuity and positivity guarantee there are no points where it suddenly drops to zero or below, reinforcing the absence of an absolute minimum.

Relative Minimum

A relative minimum is a point where the function reaches its lowest value in a specific small region around that point, even if it's not the lowest overall across the entire graph. Imagine walking along a hilly path – a relative minimum would be like a small dip or valley where you stand lower than your surroundings, but there might still be higher paths elsewhere.

• These minima are local dips in the graph.
• The function has a higher value next to this point.
• You can have multiple relative minima in a single function.

Because the function \(f(x)\) is continuous, it can fluctuate locally and give rise to relative minima. This means that there may exist points where \(f(x)\) is lower compared to immediate neighboring points, although it does not hold the lowest value globally. The example of \(f(x) = \frac{1}{x^2+1}\) shows such behavior, where local characteristics of the graph allow for these small valleys, which lead to relative minima.

Absolute Minimum

An absolute minimum is the lowest point a function reaches over its entire domain. For a function like \(f(x)\) that we’re considering, which is asymptotic to the \(x\)-axis, finding an absolute minimum is impossible. The reason is that an absolute minimum would require the function to reach a specific lowest value across all \(x\), and our function never quite meets the \(x\)-axis.

• This is the deepest point of the entire graph.
• You can only have one absolute minimum, or none at all.
• Any potential global lowest point must rise above zero in this case.

Since \(f(x) > 0\) everywhere and approaches the \(x\)-axis infinitely, there’s no single lowest point that \(f(x)\) reaches. Therefore, the concept of an absolute minimum is not applicable to such functions with horizontal asymptotes.