Problem 17 - Thomas Calculus

Question

Show that if \(f^{\prime \prime}>0\) throughout an interval \([a, b],\) then \(f^{\prime}\) has at most one zero in \([a, b] .\) What if \(f^{\prime \prime}<0\) throughout \([a, b]\) instead?

Step-by-Step Solution

Answer

Both cases result in \( f' \) having at most one zero in \([a, b]\).

Step 1: Understand the Problem

We are given a function \( f \) with its second derivative \( f'' > 0 \) on the interval \( [a, b] \). We need to show that the first derivative \( f' \) has at most one zero in this interval. We also need to consider the case where \( f'' < 0 \) and determine whether this affects \( f' \) having zeros.

Step 2: Analyze the Meaning of the Second Derivative

If \( f'' > 0 \) on \([a, b]\), the function \( f \) is concave up throughout this interval. This implies that \( f' \) is an increasing function on \([a, b]\). Conversely, if \( f'' < 0 \), then \( f \) is concave down, and \( f' \) is a decreasing function.

Step 3: Consider the Increasing Nature of an Increasing Function

Since \( f' \) is increasing when \( f'' > 0 \), the first derivative starts at a possible negative value and, moving increasing through zero, moves to positive values. This means \( f' \) can cross the x-axis only once, resulting in at most one zero on \([a, b]\).

Step 4: Consider the Decreasing Function if f'' < 0

When \( f'' < 0 \) throughout \([a, b]\), \( f' \) is a decreasing function. Therefore, \( f' \), starting from a possible positive value, can only move through zero to a negative value. This also results in at most one zero in \([a, b]\).

Step 5: Conclude the Results

In both cases, \( f' \) having at most one zero is consistent with the nature of \( f \) being either concave up or concave down on \([a, b]\). The second derivative sign controls changes in \( f' \) that leads to a single transition through zero.

Key Concepts

Second Derivative

The second derivative of a function is a crucial tool in calculus that helps us understand the behavior of functions. It is denoted by \( f'' \) and provides information about the rate of change of the function's slope. If the second derivative \( f'' \) is positive throughout a given interval, it implies that the function is "curving upwards." This behavior, known as concavity, indicates that the slope of the function (its first derivative \( f' \)) is increasing.

Conversely, if \( f'' < 0 \), the function is curving downwards. This means \( f' \) is decreasing. Ultimately, the second derivative provides insightful clues about how the shape of the graph of the function is bending. Recognizing whether a function is concave up or down is vital for predicting the potential crossings (or zeroes) of its slope.

Critical Points

Critical points are where a function's first derivative \( f' \) is zero or undefined. These points are essential in determining the behavior of the function, such as identifying potential maxima, minima, or points of inflection. When discussing critical points, it's crucial to pair this concept with the second derivative test.

For instance:

• If \( f' \) changes from negative to positive at a critical point, the point is likely a local minimum.
• If \( f' \) changes from positive to negative, it's likely a local maximum.
• If \( f' \) does not change sign, we have an inflection point if \( f'' \) also equals zero at this point.

Critical points offer the points of interest where the function's nature shifts, which is instrumental in understanding its overall behavior.

Concavity

Concavity describes the type of "curvature" of a function's graph. Specifically, if a function is concave up, it means that \( f'' > 0 \), resembling a cup opening upwards. Whereas, concave down functions have \( f'' < 0 \), forming an arch.

The concept of concavity helps in:

• Identifying whether the graph is bending towards or away from the x-axis.
• Predicting the stability of the function: concave up often indicates stability (local minimum), while down suggests potential declines (local maximum).

Concavity is a powerful tool, especially when paired with critical points, as it provides insight into whether those critical points are maxima or minima.

Increasing Function

An increasing function is simple to visualize – its output values move upwards as its input values increase. Specifically, for a function \( f \), it is increasing if \( f' > 0 \) holds true throughout the interval.

When the second derivative \( f'' > 0 \), it guarantees that \( f' \), the derivative or slope, itself is increasing. For an increasing function:

• The graph of the function rises to the right, gaining height or value.
• In terms of the derivative, \( f' \) mostly stays above or crosses the x-axis only once.

This indicates a steady upward trajectory without multiple oscillations. In our given problem, this condition helps us conclude that \( f' \) will have at most one zero.

Zero Crossing

Zero crossing refers to the point where a function's graph intersects the x-axis. For a function's derivative \( f' \), a zero crossing reflects a change in sign - moving from positive to negative or vice versa.

Recognizing zero crossings is crucial because:

• They denote moments when the function's rate of increase or decrease (\( f' \)) changes direction.
• A zero crossing can hint at a potential extremum (max or min) or indicate a flattening or leveling out.

In practical settings, a zero crossing for \( f' \) means a potential switch in the function \( f \) from increasing to decreasing, or the other way around. In our discussed problem, having \( f'' \) consistently positive or negative ensures at most one zero crossing for \( f' \), establishing the function's predictable behavior.