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The exercise you’ve shared involves analyzing and proving properties of a function \( f \), focusing on symmetry, extrema, monotonicity, and inequalities. Here’s a breakdown of the tasks:

### 1. Show that \( f \) is an odd function.
To prove this, you need to show that:
\[
f(-x) = -f(x)
\]
This means the function exhibits symmetry around the origin. The specific form of \( f \) will help determine this.

### 2. Show that \( \frac{1}{2} \) is the maximum value of \( f \) on \( [0, +\infty[ \).
Here, you're asked to demonstrate that \( f(x) \) reaches its maximum value of \( \frac{1}{2} \), likely

Montrer que f est une fonction impaire, Montrer que 1/2 est la valeur maximale de f sur [0;+∞[, Montrer que T(a,b) = (1-ab) / ((a^2+1)(b^2+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 exercise explores properties of the function f, showing it's an odd function, finding its maximum value, and analyzing monotonicity and inequalities. It also includes proving an inequality for the interval [2,4]. Suitable for advanced high school or undergraduate mathematics students.

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