Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 everywhere and has no inflection points.

Step 1: Find the First Derivative

To determine concavity, we first need the first derivative of the function. The function given is \( G(w) = w^2 - 1 \). So, compute the first derivative: \[ G'(w) = 2w \].

Step 2: Compute the Second Derivative

Now, find the second derivative, which will help us determine the concavity. The second derivative is the derivative of \( G'(w) = 2w \).\[ G''(w) = 2 \].

Step 3: Analyze the Second Derivative

The second derivative \( G''(w) = 2 \) is a constant and always positive, indicating the function is concave up for all values of \( w \).

Step 4: Identify Inflection Points

An inflection point occurs where the concavity changes, which happens when the second derivative changes sign. Since \( G''(w) = 2 \) is always positive, there are no points where the concavity changes, hence no inflection points.

Key Concepts

Second Derivative

The second derivative of a function is a powerful tool in calculus that can tell us a lot about the function's shape and behavior. It's denoted as \(G''(w)\) for our function \(G(w) = w^2 - 1\). To find the second derivative, we first need the first derivative, which is the rate of change of the function. Here, the first derivative is \(G'(w) = 2w\).
Once we have \(G'(w)\), we differentiate it again to find the second derivative. For our function, we compute:

• \(G''(w) = (2w)' = 2\)

This second derivative, \(G''(w) = 2\), is a constant number, meaning it doesn't change as \(w\) changes.
The second derivative, in this instance, tells us something important about our function's concavity without needing to test specific intervals or points.

Concavity Analysis

Concavity refers to how a function curves. If a function is concave up, it looks like a cup: \("\smile"\), and the second derivative is positive. If a function is concave down, it's like a frown: \("\frown"\), and the second derivative is negative.
For the function \(G(w) = w^2 - 1\), the second derivative is \(G''(w) = 2\). Since this is positive for all values of \(w\), the function is concave up everywhere.

• No matter what \(w\) you choose, \(G(w)\) will always have that cup-like shape.
• It means there are no intervals where the function could be concave down, as the second derivative never becomes negative.

This understanding of the function's behavior makes graphing and analyzing it much easier. Knowing \(G(w)\) is concave up gives assurance of its shape across all possible values of \(w\).

Inflection Points

Inflection points are where a function changes its concavity from up to down, or vice versa. They are special because they indicate a transition in the function's curvature. To find potential inflection points, we look for points where the second derivative equals zero or changes sign.
For the function \(G(w) = w^2 - 1\), we found \(G''(w) = 2\). This second derivative is always positive and never equals zero, indicating that there is no change in concavity throughout the entire domain.

• Since \(G''(w)\) does not change sign, there are no inflection points.
• This tells us that the curvature does not switch from concave up to concave down.

The lack of inflection points for \(G(w)\) simplifies understanding its behavior, as it remains concave up across its entire range, enhancing predictability and ease of analysis.