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The question shown in the image is:

**"2. Montrer que (a, b ∈ ℝ) : a² + b² = 1 ⇒ |a + b| ≤ √2"**

This is a mathematical proof request. The statement is saying that for any real numbers \( a \) and \( b \) such that \( a^2 + b^2 = 1 \) (which represents a point on the unit circle), we need to show that the absolute value of their sum \( |a + b| \) is less than or equal to \( \sqrt{2} \).

### Solution Outline:

We begin by interpreting the condition \( a^2 + b^2 = 1 \). This is the equation of a unit circle in the plane. Our goal is to demonstrate that the maximum value of \( |a + b| \) when \( a \) and \( b \) satisfy this equation does not exceed \( \sqrt{2} \).

1. **Rewrite the expression** \( |a + b|^2 \):
   \[
   |a + b|^2 = (a + b)^2 = a^2 + 2ab + b^2
   \]
   From \( a^2 + b^2 = 1 \), we substitute:
   \[
   |a + b|^2 = 1 + 2ab
   \]
   
2. **Find bounds for \( 2ab \)**:
   Since \( a^2 + b^2 = 1 \), both \( a \) and \( b \) lie on the unit circle, and the product \( ab \) can be maximized or minimized depending on the angle between the two points.

   Using the trigonometric parametrization of the unit circle:
   \[
   a = \cos(\theta), \quad b = \sin(\theta)
   \]
   Then:
   \[
   ab = \cos(\theta)\sin(\theta) = \frac{1}{2} \sin(2\theta)
   \]
   So, \( 2ab = \sin(2\theta) \), and the maximum value of \( \sin(2\theta) \) is 1 and the minimum is -1. Therefore:
   \[
   -1 \leq 2ab \leq 1
   \]

3. **Find bounds for \( |a + b|^2 \)**:
   Since \( |a + b|^2 = 1 + 2ab \), we substitute the bounds for \( 2ab \):
   \[
   1 - 1 \leq |a + b|^2 \leq 1 + 1
   \]
   \[
   0 \leq |a + b|^2 \leq 2
   \]

4. **Conclude**:
   Taking the square root of both sides:
   \[
   0 \leq |a + b| \leq \sqrt{2}
   \]
   Hence, we have shown that \( |a + b| \leq \sqrt{2} \), as required.

---

Would you like further clarification on any of the steps? Here are five related questions to consider:

1. How can the Cauchy-Schwarz inequality be used to derive similar bounds?
2. What are the geometric implications of this inequality on the unit circle?
3. Can this result be extended to higher dimensions, for example, with three variables?
4. What is the maximum value of \( |a + b| \) when \( a \) and \( b \) are real numbers not restricted to the unit circle?
5. How does the parametrization \( a = \cos(\theta) \) and \( b = \sin(\theta) \) relate to other trigonometric identities?

**Tip**: When working with inequalities on the unit circle, parametrization using trigonometric functions can simplify the analysis.

Montrer que (a, b ∈ ℝ) : a² + b² = 1 ⇒ |a + b| ≤ √2

Prove the inequality |a + b| ≤ √2 for real numbers a and b, given that a² + b² = 1, which represents points on the unit circle. Learn key concepts such as inequalities, trigonometric parametrization, and the Cauchy-Schwarz inequality in this detailed solution.

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