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We are given the following information:

- \( f(x) \) is positive and concave up on the interval \( I \).
- \( f''(x) \) exists, and \( g(x) = (f(x))^2 \).

We are asked to choose from a list to answer specific questions about \( f(x) \), \( g(x) \), and their derivatives on the interval \( I \). The choices are: CU (concave up), CD (concave down), \( f(x) \), \( f'(x) \), \( f''(x) \), 0, or 1.

### a) \( f''(x) > 0 \) on \( I \)

Since we know that \( f(x) \) is concave up on \( I \), this directly implies that \( f''(x) > 0 \) on \( I \).

Answer for a): **CU** (concave up).

### b) \( g'(x) = 2(A^2 + B f''(x)) \), where \( A = f'(x) \) and \( B = f(x) \)

First, we need to find \( g'(x) \), which is the derivative of \( g(x) = (f(x))^2 \).

Using the chain rule:
\[
g'(x) = 2 f(x) f'(x)
\]
So, this simplifies to \( g'(x) = 2 f(x) f'(x) \). Now, we look at the information we have:

- \( f(x) \) is positive.
- \( f'(x) \) is likely positive because \( f(x) \) is increasing if concave up.

Thus, \( g'(x) > 0 \).

Answer for b): **0**.

### c) \( g''(x) > 0 \) on \( I \)

Now, we find \( g''(x) \), which is the derivative of \( g'(x) = 2 f(x) f'(x) \).

Using the product rule:
\[
g''(x) = 2(f'(x) f'(x) + f(x) f''(x)) = 2((f'(x))^2 + f(x) f''(x))
\]
Since \( f(x) > 0 \), \( f'(x) \geq 0 \), and \( f''(x) > 0 \), we conclude that \( g''(x) > 0 \).

Answer for c): **CU** (concave up).

### d) \( g(x) \) is __ on \( I \)

Given that \( g(x) = (f(x))^2 \) and \( f(x) > 0 \), \( g(x) \) is increasing and concave up.

Answer for d): **CU** (concave up).

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### Do you need further clarification or additional details?

Here are 5 follow-up questions for practice:
1. How does concavity relate to the second derivative?
2. Can a function be concave up while its derivative is decreasing?
3. What are the conditions for \( g'(x) = 0 \)?
4. How would the behavior change if \( f(x) \) were concave down?
5. How can you determine inflection points from the second derivative?

**Tip:** When analyzing a function’s concavity, always look at the sign of the second derivative!

Suppose that on the interval I, f(x) is positive and concave up. Furthermore, assume that f''(x) exists and let g(x) = (f(x))^2. Use this information to answer the following questions:

a) f''(x) > 0 on I.

b) g'(x) = 2(A^2 + B f''(x)), where A = f'(x) and B = f(x).

c) g''(x) > 0 on I.

d) g(x) is __ on I.

Explore how to analyze a function g(x) = (f(x))^2 given f(x) is positive and concave up on an interval. This calculus problem involves derivative calculations, including the product rule and chain rule, to determine concavity and behavior of functions.

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