Let's analyze and answer each part of the question based on the information provided.

Given Information

1. $f(x)$ is positive and concave up on the interval $I$.
2. $f''(x)$ exists, and we define $g(x) = (f(x))^2$.

We are given a list of potential answers for each part:

• CU (concave up), CD (concave down)
• $f(x)$, $f'(x)$, $f''(x)$
• $0$, or $1$

Let's go through each sub-question.

Question Analysis

a) $f''(x) > \, ? \,$ on $I$.

Since $f(x)$ is concave up on $I$, it implies that $f''(x) > 0$ on $I$. Therefore:

• Answer for (a): $0$

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

For $g(x) = (f(x))^2$:

1. First derivative $g'(x) = 2f(x)f'(x)$.
2. Second derivative $g''(x) = 2(f(x)f''(x) + (f'(x))^2)$.

Comparing this with the given form $g''(x) = 2(A^2 + B f''(x))$, we identify:

• $A = f'(x)$
• $B = f(x)$

Thus:

• Answer for (b): $A = f'(x)$ and $B = f(x)$

c) $g''(x) > \, ? \,$ on $I$.

Since $f(x)$ is concave up and $f(x)$ is positive, $f''(x) > 0$. Also, since $g''(x) = 2(f(x)f''(x) + (f'(x))^2)$, and both terms are positive, we conclude $g''(x) > 0$ on $I$.

• Answer for (c): $0$

d) $g(x)$ is $?$ on $I$.

Since $g''(x) > 0$ on $I$, $g(x)$ is concave up on $I$.

• Answer for (d): CU (concave up)

Summary of Answers

• (a) 0
• (b) $A = f'(x)$ and $B = f(x)$
• (c) 0
• (d) CU

Would you like further details on any part of this solution, or do you have additional questions?

Here are five follow-up questions related to this topic:

1. How does concavity affect the interpretation of the second derivative?
2. Can we determine whether $f(x)$ is increasing or decreasing on $I$ from the given information?
3. How would the answers change if $f(x)$ were concave down instead?
4. What does the positivity of $f(x)$ imply for the function $g(x) = (f(x))^2$?
5. How would $g(x)$'s concavity be affected if $f'(x) = 0$ at some points on $I$?

Tip: When dealing with composite functions, remember to apply the chain rule carefully when differentiating.