The exercise provided involves the analysis of a mathematical function $f(x) = x - 2\sqrt{x} + 2$, alongside a sequence defined by a recurrence relation. Here's a detailed overview of the tasks:

I. Exercice 1 Breakdown

1. Domain and Limit

• Determine $D_f$, the domain of $f$.
• Calculate $\lim_{x \to +\infty} f(x)$.

2. Asymptotic Behavior

• Study the infinite branches (asymptotes) of the curve $(\mathcal{C}_f)$.

3. Relative Position

• Analyze the position of $(\mathcal{C}_f)$ relative to the line $y = x$.

4. Differentiability

• Study the differentiability of $f$ at $x = 0$.

5. Derivative and Monotonicity

• Compute $f'(x)$ and determine its behavior for $x \in [0, +\infty]$.

6. Variation Table

• Construct the variation table of $f$ (increasing/decreasing intervals).

7. Graphical Representation

• Draw the curve $(\mathcal{C}_f)$ and the line $y = x$.

Sequence Analysis

A sequence $(u_n)$ is defined by: $u_{n+1} = \frac{1}{3}u_n + n - 2, \quad u_0 = 1, \quad \forall n \in \mathbb{N}.$

8. Strictly Increasing Sequence

• Show that $(u_n)$ is strictly increasing.

9. Bounded Sequence

• Show that $(u_n)$ is unbounded.

10. Limit of the Sequence

• Determine the behavior and limits of $u_n$.

Would you like me to start solving these step by step? I can also explain concepts along the way to ensure you understand each task.

Related Questions

1. How do you determine the domain of a square root function?
2. What are asymptotes and how are they studied for a curve?
3. How do you analyze the derivative of a function to find monotonicity?
4. What methods can be used to prove a sequence is increasing or unbounded?
5. How do you find the limit of a recursive sequence?

Tip: For analyzing a function, always start with the domain, calculate limits at boundaries, and study the derivative for monotonicity.