Let's go through the problem presented in the image step by step.

Given Information:

• Let $a \in [0; 1]$.
• We define the sequence $(u_n)$ by $u_0 = a$ and for all $n \in \mathbb{N}, \ u_{n+1} = 3u_n(1 - u_n)$.

Tasks:

1. Prove that for all $x \in [0; 1]$, we have: $x(1 - x) \leq \frac{1}{4}.$

2. Prove that for all $n \in \mathbb{N}$, we have: $u_n \in [0; 1].$

Solution Outline:

1. Proving $x(1 - x) \leq \frac{1}{4}$ for $x \in [0; 1]$:

Let $f(x) = x(1 - x)$ for $x \in [0, 1]$.

• First, we calculate the derivative of $f(x)$: $f'(x) = 1 - 2x.$
• Setting $f'(x) = 0$, we find the critical point: $1 - 2x = 0 \implies x = \frac{1}{2}.$
• Now, evaluate $f(x)$ at $x = 0$, $x = 1$, and $x = \frac{1}{2}$: $f(0) = 0, \quad f(1) = 0, \quad f\left( \frac{1}{2} \right) = \frac{1}{4}.$
• Since $f(x)$ is a quadratic function opening downwards and attains its maximum value at $x = \frac{1}{2}$, the maximum value of $f(x) = x(1 - x)$ on the interval $[0, 1]$ is $\frac{1}{4}$.

Therefore, for all $x \in [0, 1]$, $x(1 - x) \leq \frac{1}{4}.$

2. Proving $u_n \in [0; 1]$ for all $n \in \mathbb{N}$:

• Base case (for $n = 0$): We are given that $u_0 = a \in [0; 1]$, so the base case holds.

• Inductive step: Assume $u_n \in [0; 1]$ for some $n \geq 0$. We need to prove that $u_{n+1} \in [0; 1]$.

  From the recurrence relation: $u_{n+1} = 3u_n(1 - u_n).$

  • If $u_n = 0$, then $u_{n+1} = 0$.
  • If $u_n = 1$, then $u_{n+1} = 0$.

  Now, consider $u_n \in (0, 1)$. We know from part 1 that for $u_n \in [0, 1]$, $u_n(1 - u_n) \leq \frac{1}{4}.$

  Therefore, $u_{n+1} = 3u_n(1 - u_n) \leq 3 \times \frac{1}{4} = \frac{3}{4}.$

  Since $u_{n+1} \geq 0$ and $u_{n+1} \leq \frac{3}{4}$, we have $u_{n+1} \in [0, 1]$.

By induction, $u_n \in [0, 1]$ for all $n \in \mathbb{N}$.

Would you like a more detailed explanation of any part? Here are some related questions that expand on this topic:

1. How can you find the global maximum of any quadratic function?
2. Can you generalize the recurrence relation for different initial conditions?
3. What is the long-term behavior of the sequence $u_n$ as $n \to \infty$?
4. Could we use other methods (e.g., graphical approaches) to verify the inequality $x(1 - x) \leq \frac{1}{4}$?
5. How does the recurrence relation change if we modify the coefficient from 3 to another constant?

Tip: When solving recurrence relations, always try to check boundary and critical points to ensure the correct behavior over an interval.