Math Problem Statement

Réponse

Soit f la fonction définie sur [9;+∞[ par f(x) = x - 2/x. Étudier sa dérivabilité, trouver sa dérivée, montrer que la droite y = x - 2 est une asymptote, étudier la position relative de f(x) et y = x - 2, et établir le tableau de variations de f.

Solution

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Question/Devoir

Énoncé :

  1. Soit ff la fonction définie sur [9;+[[9;+\infty[ par f(x)=x2xf(x) = x - \frac{2}{x}.
    a. Étudier la dérivabilité de ff sur [9;+[[9;+\infty[ puis déterminer sa dérivée f(x)f'(x).
    b. Montrer que la droite d'équation y=x2y = x - 2 est asymptote à la courbe représentative de ff dans le repère orthonormé (O;i,j)(O; \vec{i}, \vec{j}).
    c. Étudier la position relative de f(x)f(x) et y=x2y = x - 2.
    d. Établir le tableau de variations de ff.

Do you need a solution for any part of this problem or a full explanation of the derivatives, asymptotes, and table of variations? Let me know how you'd like to proceed!


Related Questions:

  1. How do you determine if a function is differentiable over an interval?
  2. What are the steps to calculate the derivative of a rational function like x2xx - \frac{2}{x}?
  3. How do you identify asymptotes of a function?
  4. How can you compare two functions to study their relative positions graphically?
  5. How do you construct a table of variations for a function?

Tip:

The asymptote y=x2y = x - 2 suggests analyzing the behavior of f(x)f(x) as x+x \to +\infty. Simplifying the terms helps to identify this asymptotic line.

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Math Problem Analysis

Mathematical Concepts

Differentiability
Derivatives
Asymptotes
Position of Functions
Function Variations

Formulas

Derivative of x → f'(x) = 1 - 2/x²
Asymptotic behavior as x → +∞: f(x) ≈ x - 2
Relative position of f(x) and y = x - 2

Theorems

Rules of Differentiation
Asymptote Determination
Behavior of Functions at Infinity

Suitable Grade Level

Grades 11-12 (High School)