https://file.aiquickdraw.com/mathbot/file/1729598688240x2l5x9d3.jpg

The image you've uploaded contains a series of exercises about a function \( f \) defined on \( \mathbb{R}^+ \) by the formula \( f(x) = x - 2\sqrt{x} + 1 \). Let me go through the questions step-by-step:

### 1) **Vérifier que \( \forall x \in \mathbb{R}^+ \), \( f(x) = (\sqrt{x} - 1)^2 \):**

We need to prove that:
\[
f(x) = x - 2\sqrt{x} + 1 = (\sqrt{x} - 1)^2
\]
By expanding the right-hand side:
\[
(\sqrt{x} - 1)^2 = (\sqrt{x})^2 - 2 \cdot \sqrt{x} \cdot 1 + 1^2 = x - 2\sqrt{x} + 1
\]
So the expression matches, and the equality is verified.

### 2) **Calculer \( \lim_{x \to +\infty} f(x) \):**

We need to compute the limit:
\[
\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty} (x - 2\sqrt{x} + 1)
\]
As \( x \to +\infty \), \( 2\sqrt{x} \) grows much slower than \( x \), so:
\[
\lim_{x \to +\infty} (x - 2\sqrt{x} + 1) = \lim_{x \to +\infty} x = +\infty
\]

### 3) **Étudier la continuité de la fonction \( f \) sur \( \mathbb{R}^+ \):**

Since the function \( f(x) = (\sqrt{x} - 1)^2 \) is a polynomial in terms of \( \sqrt{x} \), which is continuous on \( \mathbb{R}^+ \), we conclude that \( f \) is continuous on \( \mathbb{R}^+ \).

### 4) **Étudier la dérivabilité de la fonction \( f \) à droite de 0, puis interpréter géométriquement le résultat:**

First, we compute the derivative \( f'(x) \) for \( x > 0 \). Since \( f(x) = (\sqrt{x} - 1)^2 \), applying the chain rule:
\[
f'(x) = 2(\sqrt{x} - 1) \cdot \frac{1}{2\sqrt{x}} = \frac{\sqrt{x} - 1}{\sqrt{x}}
\]
Now, at \( x = 0 \), we examine the derivative from the right:
\[
f'_+(0) = \lim_{x \to 0^+} \frac{\sqrt{x} - 1}{\sqrt{x}} = -1
\]
This implies that \( f \) is differentiable at 0 from the right, with a derivative of -1, indicating that the tangent line at \( x = 0 \) has a slope of -1.

---

### Questions related to this:

1. What happens to the function \( f(x) \) as \( x \to 0^+ \)? Does it have a minimum or maximum at \( x = 0 \)?
2. How does the limit \( \lim_{x \to 0^+} f(x) \) behave?
3. Can you describe the geometric interpretation of the derivative \( f'(0) = -1 \)?
4. Is the function \( f \) strictly increasing or decreasing for all \( x \in \mathbb{R}^+ \)?
5. How does the behavior of the function \( f(x) \) change as \( x \) increases toward infinity?

### Tip: 
When checking for continuity or differentiability at a point, it’s useful to verify both left-hand and right-hand limits and derivatives to ensure smoothness or detect possible corner points.

Soit f la fonction définie sur R+ par : f(x) = x - 2√x + 1. Vérifier que (∀x ∈ IR+): f(x) = (√x - 1)^2, Calculer lim f(x) x→+∞, Étudier la continuité de la fonction f sur IR+, Étudier la dérivabilité de la fonction f à droite de 0, etc.

This exercise explores the function f(x) = x - 2√x + 1 defined on R+. It involves verifying equality with (√x - 1)^2, calculating limits, studying continuity and differentiability, and finding the equation of a tangent line. A great problem for university-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation