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?

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.