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.