Math Problem Statement

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.

Solution

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

1. Show that ff is an odd function.

To prove this, you need to show that: f(−x)=−f(x)f(-x) = -f(x) This means the function exhibits symmetry around the origin. The specific form of ff will help determine this.

2. Show that 12\frac{1}{2} is the maximum value of ff on [0,+∞[[0, +\infty[.

Here, you're asked to demonstrate that f(x)f(x) reaches its maximum value of 12\frac{1}{2}, likely

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Math Problem Analysis

Mathematical Concepts

Odd Function
Maxima and Minima
Inequalities
Monotonicity
Variation Table

Formulas

f(-x) = -f(x) for odd functions
1/2 as maximum of f on [0, +∞[
T(a,b) = (1-ab) / ((a^2+1)(b^2+1))
Monotonicity conditions on [0,1] and [1,+∞[
Inequality 5/2 ≤ x + 1/x ≤ 17/4

Theorems

Properties of Odd Functions
Monotonicity Theorem
Critical Points for Maxima and Minima
Inequality Theorems

Suitable Grade Level

Undergraduate Mathematics or Advanced High School

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