Problem 1 - Calculus

Question

Fill in the blanks: (a) If \(f^{\prime \prime}\) is positive on an interval, then \(f^{\prime}\) is _________ on that interval, and \(f\) is ___________ on that interval. (b) If \(f^{\prime \prime}\) is negative on an interval, then \(f^{\prime}\) is __________ on that interval, and \(f\) is __________ on that interval.

Step-by-Step Solution

Answer

(a) increasing, concave up; (b) decreasing, concave down.

Step 1: Understanding Concavity and Derivatives

To solve this problem, we need to understand how the second derivative, \(f''\), influences \(f'\) and \(f\). The second derivative provides information about the concavity of the function \(f\).

Step 2: Analyzing Positive Second Derivative

When \(f''\) is positive on an interval, it indicates that the function \(f\) is concave up on that interval. As a result of concave up, the first derivative \(f'\) is increasing.

Step 3: Conclusion for Positive Second Derivative

Thus, when \(f''\) is positive, \(f'\) increases and \(f\) is concave up.

Step 4: Analyzing Negative Second Derivative

When \(f''\) is negative on an interval, it indicates that the function \(f\) is concave down on that interval. As a result of concave down, the first derivative \(f'\) is decreasing.

Step 5: Conclusion for Negative Second Derivative

Thus, when \(f''\) is negative, \(f'\) decreases and \(f\) is concave down.

Key Concepts

Second Derivative

The second derivative of a function, denoted as \( f'' \), plays a crucial role in understanding the behavior of a function. It provides insights into the curvature and concavity of the graph of the function. Essentially, the second derivative tells us how the rate of change of the function's slope (first derivative) is itself changing. If the second derivative is positive, it indicates that the slope of the function is increasing over the interval. Conversely, if the second derivative is negative, it means the slope is decreasing. By analyzing the sign of \( f'' \), we can deduce important information about whether the function is concave up or concave down. This helps in sketching graphs and in understanding the nature of the function in more detail.

Concave Up

When a function is described as 'concave up' on an interval, it means that the graph of the function bends upwards, forming a U-like shape. This occurs when the second derivative \( f'' \) is positive on that interval. A useful way to visualize this is to think of the graph as a bowl that can hold water. In mathematical terms, this happens when \( f'' > 0 \).
This positive second derivative implies that the first derivative \( f' \) is increasing over this interval, meaning the slope of the tangent lines to \( f \) are getting steeper.
Key points to remember about concave up:

• The function rises faster as you move along the x-axis.
• The slope (first derivative) of the function is increasing.
• The second derivative is positive.

Concave Down

A function is 'concave down' on an interval if the graph bends or curves downwards, often resembling an upside-down U or an arch. This situation arises when the second derivative \( f'' \) is negative on that interval. It indicates that the function's graph cannot "hold water" as it forms a kind of dome shape.
When \( f'' < 0 \), the rate of change of the slope decreases. This means that the first derivative \( f' \) is decreasing, making the tangent lines less steep.
Key points about concave down include:

• The function is slowing down its rise or possibly beginning to descend.
• The slope (first derivative) of the function decreases.
• The second derivative is negative.

Increasing Function

An increasing function is one where as you move from left to right on the graph (from lower values of \( x \) to higher values), the values of the function also increase. For the graph of a function \( f \), this is typically characterized by an upward slant. If the first derivative \( f' \) is positive on an interval, this means the function is constantly rising in that interval.
Some key aspects of increasing functions:

• The values of \( f(x) \) become larger as \( x \) gets larger.
• The slopes of tangent lines (or the first derivative) remain positive.
• In the context where \( f'' > 0 \), even the rate at which \( f \) increases is increasing.

Decreasing Function

Conversely, a decreasing function is one where, as you move left to right across the graph, the function values fall. This is depicted by a downward direction in the graph's slope. When the first derivative \( f' \) is negative on an interval, this signals that the function is decreasing throughout that interval.
Characteristics of a decreasing function include:

• The function values \( f(x) \) reduce as \( x \) goes up.
• The slopes of tangent lines (or the first derivative) are negative, indicating a downward trend.
• If \( f'' < 0 \), the rate at which \( f \) decreases is also increasing.