https://file.aiquickdraw.com/mathbot/file/1725720575069gxkggjxz.jpg

The image contains the mathematical function:

\[
f(x) = 2x + \frac{2}{x}, \quad x \in \left[ \frac{1}{2}, 2 \right]
\]

This expression defines a function \( f(x) \) with the domain restricted to the interval \( \left[ \frac{1}{2}, 2 \right] \). The function combines a linear term \( 2x \) and a rational term \( \frac{2}{x} \).

Would you like to explore this function further, such as calculating its derivative, critical points, or behavior within the given interval?

Here are five related questions:
1. What is the derivative of \( f(x) \) and where are its critical points?
2. What is the behavior of \( f(x) \) as \( x \) approaches the endpoints of the interval?
3. Is \( f(x) \) increasing or decreasing in different parts of the interval \( \left[ \frac{1}{2}, 2 \right] \)?
4. Can we find the global minimum or maximum of \( f(x) \) within this interval?
5. How does the second derivative \( f''(x) \) behave, and what does it say about the concavity of the function?

**Tip:** To analyze the behavior of a function on a given interval, checking the first and second derivatives can provide insights into increasing/decreasing behavior and concavity.

Analyze the function f(x) = 2x + 2/x, where x belongs to the interval [1/2, 2].

Explore the properties of the function f(x) = 2x + 2/x within the interval [1/2, 2]. This analysis includes the calculation of the first and second derivatives, critical points, and behavior of the function using calculus concepts.

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