This image presents a mathematical exercise for a 2Bac level involving functions and sequences. Below is a structured response to help understand and solve the problems:

1. Understanding the Problem:

• A function $f(x) = x - 2\sqrt{x} + 2$ is provided.

• The goal is to analyze its domain, limits, derivatives, asymptotes, and graph.

• A sequence $(u_n)$ defined as $u_{n+1} = \frac{1}{3}u_n + n - 2$ is also introduced, where $u_0 = 1$.

2. Step-by-Step Solutions for Each Question:

Function Analysis $f(x) = x - 2\sqrt{x} + 2$:

1. Domain $D_f$ and Limit $\lim_{x \to +\infty} f(x)$:

   • $D_f = [0, +\infty[$ because the square root $\sqrt{x}$ requires non-negative $x$.
   • For $\lim_{x \to +\infty} f(x)$: f(x) = x - 2\sqrt{x} + 2 \quad \implies \quad \lim_{x \to +\infty} f(x) = +\infty \, \text{(since x dominates)}.

2. Behavior at Infinity (Branches):

   • Since $x \to +\infty$, the $x$-term dominates. The function increases indefinitely.

3. Position Relative to $y = x$:

   • The straight line $y = x$ serves as a reference. Comparing $f(x)$ and $y = x$: $f(x) - x = -2\sqrt{x} + 2.$ For large $x$, $-2\sqrt{x} \to -\infty$, so $f(x) < y$.

4. Derivability and Derivative:

   • $f(x)$ is derivable on $]0, +\infty[$ because the square root function is smooth.
   • Derivative: $f'(x) = 1 - \frac{1}{\sqrt{x}}.$

5. Expression of $f'(x)$:

   • As stated: $f'(x) = 1 - \frac{1}{\sqrt{x}}$, valid for $x \in ]0, +\infty[$.

6. Table of Variation:

   • Analyze the sign of $f'(x)$:
     • $f'(x) = 0$ when $1 - \frac{1}{\sqrt{x}} = 0 \implies \sqrt{x} = 1 \implies x = 1$.
     • For $x < 1$, $f'(x) < 0$ (decreasing).
     • For $x > 1$, $f'(x) > 0$ (increasing).
   • Conclusion: $f(x)$ decreases on $[0, 1]$ and increases on $[1, +\infty[$.

7. Construction of Graph $(C_f)$ and Line $(\Delta)$:

   • $(C_f)$ is drawn based on the increasing/decreasing behavior and compared with $y = x$.

Sequence $u_n$:

1. Recurrence $u_{n+1} = \frac{1}{3}u_n + n - 2$:

   • Starting at $u_0 = 1$.

2. Proving $u_n > 1$:

   • Use induction to prove that $u_n > 1$ for all $n \in \mathbb{N}$.

3. Showing the Sequence is Decreasing:

   • Analyze $u_{n+1} - u_n$ to check if it is negative.

4. Convergence of $u_n$:

   • Use limits to determine whether the sequence converges and find the limit.

3. Do You Want Detailed Solutions?

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4. Related Questions:

1. What happens if we change $f(x)$ to $x - 3\sqrt{x} + 5$?
2. How do we find asymptotes of a function analytically?
3. Prove the convergence of a sequence given a recurrence relation.
4. Can $f(x) = x - 2\sqrt{x} + 2$ have a minimum or maximum value?
5. What are the graphical properties of functions involving square roots?

Tip:

When analyzing a function, always start with the domain, then move to limits and derivatives to understand its overall behavior.