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The exercise consists of several problems concerning functions \( f \) and \( g \) defined on \( \mathbb{R} \), with specific expressions given for \( f(x) = x^2 - 2x + 1 \) and \( g(x) = -2x^2 + 4x + 1 \). Let's go through each part step by step.

### Problem Breakdown:

#### 1. Part (a)
   - **Task:** Compare \( f(x) \) and \( g(x) \) across specific intervals: \( ]-\infty; 0] \), \( ]2; +\infty[ \), and \( [0; 2] \).
   - **Approach:** To compare \( f(x) \) and \( g(x) \), we can analyze their behavior over each interval by finding points of intersection and determining if one function is greater than the other over the interval.

#### 2. Part (b)
   - **Task:** Deduce the relative positions of \( f \) and \( g \) over the intervals \( ]-\infty; 0] \), \( ]2; +\infty[ \), and \( [0; 2] \).
   - **Approach:** This step likely builds on the previous part, requiring us to specify which function lies above or below the other across these intervals.

#### 3. Graph Interpretation Problem
   - **Task:** Analyze the graph of \( f(x) \) and \( g(x) \) to solve inequalities involving \( f(x) \) and \( g(x) \), such as:
     - \( f(x) \leq g(x) \)
     - \( f(x) > g(x) \)
     - \( f(x) \geq 0 \)
     - \( f(x) < 0 \)
     - \( f(x) = g(x) \)

   - **Approach:** From the graph, identify the regions where each inequality is satisfied based on where one curve lies above or below the other.

#### 4. Table of Variations for \( f(x) \)
   - **Task:** Use the table of variations provided to determine values such as \( f([-2;4]) \), \( f([4;8]) \), \( f([-7;4]) \), and \( f([-7;8]) \).
   - **Approach:** This involves reading values or ranges of \( f(x) \) over the specified intervals by using the increasing or decreasing behavior shown in the variation table.

#### 5. Graph Analysis for a Function Defined on \( I = [-3;4] \)
   - **Task:** Analyze the behavior of a function defined on this interval by examining its graph.
   - **Approach:** Interpret the graph for details like the function’s range, critical points, and behavior (increasing or decreasing) over the interval.

### Let’s Start with Solutions:

1. **For part (a)**: We’ll compare \( f(x) = x^2 - 2x + 1 \) and \( g(x) = -2x^2 + 4x + 1 \) over each interval.
   - To proceed, we can set \( f(x) = g(x) \) to find intersections and analyze which function is greater within the intervals.
2. **For part (b)**: This step will directly follow from the conclusions reached in part (a).
3. **Graph Interpretation**: Reading the points of intersection and inequalities from the graph.
4. **Variation Table Analysis**: Using the table to find specified values of \( f(x) \) over given intervals.
5. **Final Graph Analysis**: Observing the function behavior on \( I = [-3;4] \).

Would you like me to go through each solution in more detail? 

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### Additional Related Questions:
1. What are the steps to determine where \( f(x) = g(x) \) algebraically?
2. How do we analyze the behavior of functions over intervals?
3. What does the variation table tell us about a function’s growth?
4. How can we deduce inequalities graphically for two functions?
5. What methods can we use to identify the intervals where one function is greater than another?

### Tip:
When comparing functions, finding points of intersection helps identify where one function transitions from being greater to less than another across intervals.

Soient f et g deux fonctions définies sur R par f(x) = x^2 - 2x + 1 et g(x) = -2x^2 + 4x + 1. Comparez f et g pour tout x dans les intervalles suivants : ]-∞;0], ]2;+∞[ et [0;2]. Déduire les positions relatives sur ]-∞;0], ]2;+∞[ et [0;2]. Résoudre graphiquement : f(x) = 0, g(x) = 2, g(x) ≤ 2, f(x) ≤ g(x), f(x) > g(x), f(x) ≥ 0, f(x) < 0, f(x) = g(x). Soit f une fonction numérique dont le tableau de variations est le suivant : x = [-7, 8], f([-2, 4]), f([4, 8]), f([-7, 4]), et f([-7, 8]).

Explore the comparison of functions f(x) = x^2 - 2x + 1 and g(x) = -2x^2 + 4x + 1 over different intervals. Solve inequalities graphically and deduce positions of the functions. Analyze a table of variations and understand the behavior of functions. Suitable for high school students in Grades 10-12.

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