https://r2.erweima.ai/mathbot/image/1724685993528nbkzhbnv.jpg

To solve the problem presented in the image, let's analyze it step by step.

### Problem Statement:
The question is asking at what value of \( x \) the function \( f(x) \) will have a global minimum based on the graph and data points provided.

### Analysis of the Graph:
1. The graph appears to be a parabolic curve, which is typical for quadratic functions.
2. The function \( f(x) \) reaches its minimum point at the vertex of the parabola.

### Observations:
- The curve shows a gradient descent path towards the minimum, indicated by small circles and a decreasing line.
- The green circle represents the "Optimal Point (GD)," which is the point of global minimum achieved by gradient descent.

### Conclusion:
- The global minimum of the function \( f(x) \) occurs at the point where the vertex of the parabola is located.
- Based on the graph, the vertex (and thus the global minimum) is at \( x = 2 \).

### Final Answer:
The global minimum value for \( f(x) \) occurs at \( x = 2 \).

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Would you like more detailed explanations, or do you have any other questions?

Here are five additional questions related to this topic:
1. What is the general form of a quadratic function, and how do you determine its vertex?
2. How does gradient descent work, and why is it used to find the minimum of a function?
3. What are the conditions for a point to be a global minimum in a function?
4. How does the choice of learning rate affect the gradient descent process?
5. Can gradient descent converge to a local minimum instead of a global minimum? Under what circumstances?

**Tip:** When analyzing graphs of functions, always look for key features like intercepts, vertices, and points of inflection to understand the behavior of the function.

Learn how to find the global minimum of the quadratic function f(x) through detailed analysis and graph interpretation. This problem involves calculus concepts such as optimization and the vertex form of quadratic functions. Suitable for high school students.

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