3. For $n \geq 2$, let $O_{n-1} = \{(x_1, \ldots, x_{n-1}) \in \mathbb{R}^{n-1} \mid 0 < x_1 < x_2 < \cdots < x_{n-1}\}$ and let $U_{n-1}$ be the set of $(n-1)$-tuples $(y_1, \ldots, y_{n-1}) \in \mathbb{R}^{n-1}$ such that $$0 < y_1, \quad y_i > y_{i+1} \text{ if } i \in \{1, \ldots, n-2\} \text{ is odd}, \quad y_i < y_{i+1} \text{ if } i \in \{1, \ldots, n-2\} \text{ is even}.$$ For $x \in O_{n-1}$, we define the function $\pi_x \in \mathscr{P}_n$ by $\pi_x(t) = t(x_1 - t) \cdots (x_{n-1} - t)$. We define the map $Y = (Y_1, \ldots, Y_{n-1}) : O_{n-1} \rightarrow \mathbb{R}^{n-1}$ by $$Y_i(x) = \int_0^{x_i} \pi_x(u)\, du, \quad x = (x_1, \ldots, x_{n-1}) \in O_{n-1}$$ a. Let $j \in \{1, \ldots, n-1\}$ and $x \in O_{n-1}$. Show that $$d_{x,j} : t \mapsto \int_0^t u \prod_{1 \leq \ell \leq n-1, \ell \neq j} (x_\ell - u)\, du$$ is in $\mathscr{P}_n$ and vanishes with its derivative at 0. Deduce the existence of $\chi_{x,j} \in \mathscr{P}_{n-2}$ satisfying $$\forall t \in \mathbb{R}, \quad d_{x,j}(t) = t^2 \chi_{x,j}(t).$$ b. For $x \in O_{n-1}$ and $(i,j) \in \{1, \ldots, n-1\}^2$, show the existence of $\frac{\partial Y_i}{\partial x_j}(x)$ and verify that $$\frac{\partial Y_i}{\partial x_j}(x) = d_{x,j}(x_i)$$ Deduce that $Y$ is a map of class $C^1$ on the open set $O_{n-1}$, with values in $U_{n-1}$. c. Prove that for $x \in O_{n-1}$, the set $\{\chi_{x,j} \mid j \in \{1, \ldots, n-1\}\}$ is a basis of $\mathscr{P}_{n-2}$. d. Deduce that the differential of $Y$ at point $x$ is invertible.
3. For $n \geq 2$, let $O_{n-1} = \{(x_1, \ldots, x_{n-1}) \in \mathbb{R}^{n-1} \mid 0 < x_1 < x_2 < \cdots < x_{n-1}\}$ and let $U_{n-1}$ be the set of $(n-1)$-tuples $(y_1, \ldots, y_{n-1}) \in \mathbb{R}^{n-1}$ such that
$$0 < y_1, \quad y_i > y_{i+1} \text{ if } i \in \{1, \ldots, n-2\} \text{ is odd}, \quad y_i < y_{i+1} \text{ if } i \in \{1, \ldots, n-2\} \text{ is even}.$$
For $x \in O_{n-1}$, we define the function $\pi_x \in \mathscr{P}_n$ by $\pi_x(t) = t(x_1 - t) \cdots (x_{n-1} - t)$. We define the map $Y = (Y_1, \ldots, Y_{n-1}) : O_{n-1} \rightarrow \mathbb{R}^{n-1}$ by
$$Y_i(x) = \int_0^{x_i} \pi_x(u)\, du, \quad x = (x_1, \ldots, x_{n-1}) \in O_{n-1}$$
a. Let $j \in \{1, \ldots, n-1\}$ and $x \in O_{n-1}$. Show that
$$d_{x,j} : t \mapsto \int_0^t u \prod_{1 \leq \ell \leq n-1, \ell \neq j} (x_\ell - u)\, du$$
is in $\mathscr{P}_n$ and vanishes with its derivative at 0. Deduce the existence of $\chi_{x,j} \in \mathscr{P}_{n-2}$ satisfying
$$\forall t \in \mathbb{R}, \quad d_{x,j}(t) = t^2 \chi_{x,j}(t).$$
b. For $x \in O_{n-1}$ and $(i,j) \in \{1, \ldots, n-1\}^2$, show the existence of $\frac{\partial Y_i}{\partial x_j}(x)$ and verify that
$$\frac{\partial Y_i}{\partial x_j}(x) = d_{x,j}(x_i)$$
Deduce that $Y$ is a map of class $C^1$ on the open set $O_{n-1}$, with values in $U_{n-1}$.\\
c. Prove that for $x \in O_{n-1}$, the set $\{\chi_{x,j} \mid j \in \{1, \ldots, n-1\}\}$ is a basis of $\mathscr{P}_{n-2}$.\\
d. Deduce that the differential of $Y$ at point $x$ is invertible.