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The problem provides the following:

- \( f(x) \) is a continuous function on \( \mathbb{R} \), and its derivative, \( f'(x) \), changes sign according to the table provided.
- The function \( f(x) \) has critical points at \( x = -1, x = 0, \) and \( x = 1 \).
- The goal is to find the minimum value of the function \( g(x) = 3x^2 + 2025 + 2024f(x) \) on the interval \( [1, 2] \).

### Analyzing the given dataThe problem provides the following:

- \( f(x) \) is a continuous function on \( \mathbb{R} \), and its derivative, \( f'(x) \), changes sign according to the table provided.
- The function \( f(x) \) has critical points at \( x = -1, x = 0, \) and \( x = 1 \).
- The goal is to find the minimum value of the function \( g(x) = 3x^2 + 2025 + 2024f(x) \) on the interval \( [1, 2] \).

### Analyzing the given 
From the table, we see:
- At \( x = 1 \), \( f(x) = 1 \) (local minimum).
- At \( x = 2 \), \( f(x) \to +\infty \) (since the function is increasing after \( x = 1 \)).

### Steps:
1. We evaluate \( g(x) = 3x^2 + 2025 + 2024f(x) \) at \( x = 1 \) and \( x = 2 \).
   
At \( x = 1 \):
\[
g(1) = 3(1)^2 + 2025 + 2024 \cdot f(1) = 3 + 2025 + 2024 \cdot 1 = 3 + 2025 + 2024 = 4052
\]

As \( x \to 2 \), since \( f(x) \to +\infty \), we can conclude that \( g(x) \) tends to infinity.

Thus, the minimum value of \( g(x) \) on the interval \( [1, 2] \) is at \( x = 1 \), which gives:
\[
\boxed{4052}
\]

Would you like more details on the reasoning? Here are some related questions to deepen your understanding:

1. Why does the value of \( g(x) \) increase as \( x \to 2 \)?
2. How do we use the first derivative test to identify critical points of \( f(x) \)?
3. What is the significance of the behavior of \( f'(x) \) between the critical points?
4. How does the continuity of \( f(x) \) help in solving the problem?
5. Could we use the second derivative to check concavity for \( f(x) \)?

**Tip**: Always examine endpoints when looking for minimum or maximum values of a function on a closed interval.

Cho hàm số f(x) liên tục trên ℝ và có bảng biến thiên như hình bên dưới. Tìm giá trị nhỏ nhất của hàm số g(x) = 3x^2 + 2025 + 2024f(x) trên đoạn [1; 2].

Giải bài toán tìm giá trị nhỏ nhất của hàm số g(x) = 3x^2 + 2025 + 2024f(x) dựa trên bảng biến thiên của hàm số f(x) và các điểm tới hạn trên đoạn [1, 2]. Áp dụng phép thử đạo hàm bậc nhất để xác định điểm cực trị.

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