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I see you've uploaded an image containing several math problems, written in French. Here's a translation and breakdown of the problems presented:

1. **Montrer que \( f \) est une fonction impaire.**
   - Show that \( f \) is an odd function.

2. **Montrer que \( \frac{1}{2} \) est la valeur maximale de \( f \) sur \( [0; +\infty[ \).**
   - Show that \( \frac{1}{2} \) is the maximum value of \( f \) on the interval \( [0; +\infty[ \).

3. **Soient \( a \) et \( b \) deux éléments distincts de \( \mathbb{R}^+ \), montrer que :**
   \[
   T(a, b) = \frac{1 - ab}{(a^2 + 1)(b^2 + 1)}.
   \]
   - Let \( a \) and \( b \) be two distinct elements of \( \mathbb{R}^+ \) (the set of positive real numbers), show that:
   \[
   T(a, b) = \frac{1 - ab}{(a^2 + 1)(b^2 + 1)}.
   \]

4. **Montrer que \( f \) est strictement croissante sur \( [0, 1] \) et strictement décroissante sur \( [1, +\infty[ \).**
   - Show that \( f \) is strictly increasing on \( [0, 1] \) and strictly decreasing on \( [1, +\infty[ \).

5. **Dresser le tableau de variations de \( f \) sur \( \mathbb{R} \).**
   - Draw the variation table of \( f \) on \( \mathbb{R} \) (the set of real numbers).

6. **En déduire que pour tout \( x \) de l'intervalle \( [2, 4] \), on a :**
   \[
   \frac{5}{2} \leq x + \frac{1}{x} \leq \frac{17}{4}.
   \]
   - Deduce that for every \( x \) in the interval \( [2, 4] \), we have:
   \[
   \frac{5}{2} \leq x + \frac{1}{x} \leq \frac{17}{4}.
   \]

Would you like me to work through one of these problems, or clarify any specific part?

Here are some follow-up questions for you to consider:

1. Do you want a detailed explanation of how to prove \( f \) is an odd function?
2. Should we explore the maximum value of \( f \) over the specified interval?
3. Would you like help understanding the behavior of the function \( T(a, b) \)?
4. Should we discuss why \( f \) is strictly increasing and decreasing over certain intervals?
5. Would you like me to construct the variation table for \( f \)?

**Tip:** When tackling problems involving function variations, always start by checking the first derivative to identify critical points and intervals of monotonicity.

Montrer que f est une fonction impaire. Montrer que 1/2 est la valeur maximale de f sur [0; +∞[. Soient a et b deux éléments distincts de R+, montrer que T(a,b) = (1 - ab) / ((a² + 1)(b² + 1)). Montrer que f est strictement croissante sur [0, 1] et strictement décroissante sur [1, +∞[. Dresser le tableau de variations de f sur R. En déduire que pour tout x de l'intervalle [2, 4], on a: 5/2 ≤ x + 1/x ≤ 17/4.

This problem explores key concepts in real analysis, including proving that a function is odd, finding the maximum value of f, and analyzing its monotonicity over specified intervals. Additionally, students are tasked with working through inequalities in the interval [2, 4].

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