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.