Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-18, use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. \(G(w)=w^{2}-1\)

Step-by-Step Solution

Answer

The function is concave up for all \( w \), with no inflection points.

Step 1: Find the First Derivative

To begin analyzing the concavity of the function, we need to determine the first derivative of the function. The function given is \( G(w) = w^2 - 1 \). The first derivative \( G'(w) \) is calculated as follows: \( G'(w) = 2w \).

Step 2: Find the Second Derivative

The second step involves finding the second derivative of the function. This will help us use the Concavity Theorem. From the first derivative \( G'(w) = 2w \), the second derivative is \( G''(w) = 2 \).

Step 3: Analyze the Second Derivative

Using the Concavity Theorem, we analyze the sign of the second derivative to determine the concavity. Since \( G''(w) = 2 \) is positive for all \( w \), this implies the function \( G(w) \) is concave up for all \( w \).

Step 4: Identify Inflection Points

An inflection point occurs where the second derivative changes sign. However, since \( G''(w) = 2 \) does not change sign and remains positive, there are no inflection points in the function \( G(w) = w^2 - 1 \).

Key Concepts

First Derivative

The first derivative is a crucial concept in calculus. It provides the rate of change of the function concerning its variable.
In the given exercise, we have the function \( G(w) = w^2 - 1 \). To find the first derivative, you differentiate it with respect to \( w \).

• The first derivative \( G'(w) = 2w \) indicates how fast or slow the function \( G(w) \) is changing at any point \( w \).

This derivative tells us something about the slope of the tangent line to the curve at any point.

• If \( G'(w) > 0 \), the function is increasing at that point.
• If \( G'(w) < 0 \), the function is decreasing at that point.

By finding \( G'(w) \), we lay the groundwork for later steps in analyzing the behavior of the function, including determining concavity.

Second Derivative

The second derivative helps delve deeper into the behavior of a function by analyzing the rate of change of the first derivative.
For our function \( G(w) = w^2 - 1 \), we already found the first derivative as \( G'(w) = 2w \). The second derivative \( G''(w) = 2 \) further analyzes concavity.

• A constant positive second derivative, like \( G''(w) = 2 \), indicates that the original function is always concave up.
• A constant negative second derivative indicates concave down.
• If the second derivative were zero or changed sign at any point, that would identify a point of potential inflection.

Thus, since \( G''(w) \) remains constant and positive, \( G(w) \) is concave up across its entire domain.

Inflection Points

Inflection points are points on a curve where the concavity changes from concave up to concave down or vice versa. This requires that the second derivative must change its sign.
However, with \( G(w) = w^2 - 1 \), the second derivative is \( G''(w) = 2 \) which is positive and does not change sign.

• This means there are no inflection points as the function constantly maintains its concavity.

Identifying inflection points is crucial as they provide insights into the structure of the graph of the function, particularly how the slopes transition across different segments.