The image contains a mathematical problem where a function $f(x)$ is defined as:

$f(x) = \lim_{n \to \infty} \prod_{n=0}^{\infty} \left( \frac{x^{2^n} + 1}{x^{2^{n+1}} + 1} \right)^2$

The task is to match items from List-I with corresponding intervals from List-II. Here's the list:

List-I:

• (P) $f(x)$ is increasing in
• (Q) $f(x)$ is decreasing in
• (R) $f(x)$ has a local minimum in
• (S) $f(x) = x$ has only one solution in

List-II:

• (1) $x \in (1, \infty)$
• (2) $x \in (0, \frac{1}{2}) \cup (5, \infty)$
• (3) $x \in (0, 1)$
• (4) $x \in (0, 2)$
• (5) $x \in (0, 1) \cup (4, 5)$

The options (A), (B), (C), and (D) provide different mappings between these lists.

The user has partially matched the following:

• $P$ corresponds to $(1)$
• $Q$ corresponds to $(2)$

Would you like a detailed explanation of how to approach solving this problem and confirm these matches or continue exploring other parts of the solution?

5 Related Questions:

1. What is the general behavior of the given infinite product for $f(x)$?
2. How can you determine whether $f(x)$ is increasing or decreasing in specific intervals?
3. What are the conditions for $f(x)$ to have a local minimum in an interval?
4. How do you solve the equation $f(x) = x$ in various intervals?
5. What does the limit and infinite product suggest about the global behavior of $f(x)$?

Tip:

For problems involving infinite products and limits, analyzing the behavior of partial products and considering specific test values for $x$ can help understand the function's behavior in different intervals.