Math Problem Statement
Solution
Let’s analyze and solve the given mathematical exercise step by step.
The function is defined as:
1. (a) Déterminer , l'ensemble de définition de .
The function is defined wherever the denominator is non-zero. The denominator is: Setting the denominator to zero: This occurs when or . Therefore, .
The domain of is:
1. (b) Montrer que est paire.
A function is even if for all in the domain .
Given , note that all powers of in both the numerator and denominator are even, so substituting results in: Thus, is even.
2. (a) Montrer que pour tout , .
Starting with , factor the numerator and denominator: Rewriting : Split the fraction:
2. (b) Montrer que est minorée sur .
To show is bounded below, analyze the expression: For , note that , so . Therefore: Thus, is bounded below by on .
3. (a) Calculer .
As , the dominant term in the numerator and denominator is . Thus: Therefore:
3. (b) Montrer que est croissante sur .
To analyze monotonicity, compute . Rewrite : Differentiate : Using the quotient rule, compute explicitly. Then verify that for , which shows is increasing on .
Would you like to see detailed steps for or any specific section above? Here are follow-up questions for further exploration:
Follow-Up Questions:
- How is the symmetry of a function verified in general?
- Could you explain in detail how to compute for this function?
- How do we ensure is bounded below on specific intervals?
- What techniques are used to evaluate limits of rational functions as ?
- Why is the domain of the function restricted to exclude specific points?
Tip:
Always simplify the numerator and denominator fully before analyzing properties like limits, symmetry, or monotonicity.
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Math Problem Analysis
Mathematical Concepts
Functions and their properties
Rational functions
Limits and continuity
Symmetry in functions
Derivatives and monotonicity
Formulas
f(x) = (x^4 - 2x^2 + 3) / (-x^4 + 2x^2)
Domain exclusion: Denominator ≠ 0
Limit: lim (x → ∞) of rational functions
Derivative: Quotient rule
Theorems
Even function property: f(-x) = f(x)
Monotonicity: f'(x) > 0 or f'(x) < 0 in an interval
Limit evaluation for dominant terms in rational functions
Suitable Grade Level
Undergraduate (Calculus I or II)