Let $U$ be the open set of $\mathbb{R}^{2}$ defined by $$U := \left\{(t, x) \in \mathbb{R}^{2} \mid t > 0 \text{ and } x > -t\right\}$$ and let $f$ be the function defined on $U$ by $$f(t, x) = t^{2} \ln\left(1 + \frac{x}{t}\right) - tx.$$
(a) Show that for $(t, x) \in U$, we have $$x \leqslant 0 \Rightarrow f(t, x) \leqslant -\frac{x^{2}}{2}.$$
(b) For $x > 0$, show that we have $$\forall t \geqslant 1, \quad f(t, x) \leqslant f(1, x).$$ For this, one may begin by writing $\frac{\partial f}{\partial t}(t, x)$ in the form $t F(x/t)$ for a certain function $F$ that one will study.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f: U \to \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that for all $x \in U$, $\Delta f(x) > 0$. Let $x_0 \in \bar{U}$ be a point where $f$ attains its maximum. Show that $x_0 \in \partial U$ and deduce that $\forall x \in U, f(x) < \sup_{y \in \partial U} f(y)$.
One may assume by contradiction that $x_0 \in U$, justify that there exists $i \in \llbracket 1, n \rrbracket$ such that $\frac{\partial^2 f}{\partial x_i^2}(x_0) > 0$, and consider the function $\varphi$ defined, for $t$ real, by $\varphi(t) = f(x_0 + t e_i)$, where $e_i$ denotes the $i$-th vector of the canonical basis of $\mathbb{R}^n$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Show that $g_\varepsilon$ is a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that $\forall x \in U, \Delta g_\varepsilon(x) > 0$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Deduce that $\forall x \in U, f(x) \leqslant \sup_{y \in \partial U} f(y)$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f : U \rightarrow \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that for all $x \in U$, $\Delta f(x) > 0$. Show that $x_0 \in \partial U$ and deduce that $\forall x \in U, f(x) < \sup_{y \in \partial U} f(y)$.
One may assume by contradiction that $x_0 \in U$, justify that there exists $i \in \llbracket 1, n \rrbracket$ such that $\frac{\partial^2 f}{\partial x_i^2}(x_0) > 0$, and consider the function $\varphi$ defined, for $t$ real, by $\varphi(t) = f(x_0 + t e_i)$, where $e_i$ denotes the $i$-th vector of the canonical basis of $\mathbb{R}^n$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Show that $g_\varepsilon$ is a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that $\forall x \in U, \Delta g_\varepsilon(x) > 0$.

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $h _ { n }$ the map from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad h _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \cdots + x _ { n } - 1 .$$
We admit that $F _ { n }$ and $h _ { n }$ are both of class $\mathcal { C } ^ { 1 }$ on $U _ { n }$. We denote by $H _ { n }$ the set $H _ { n } = \left\{ \left( x _ { 1 } , \ldots , x _ { n } \right) \in \mathbb { R } ^ { n } \mid x _ { 1 } + \cdots + x _ { n } = 1 \right\}$.
Determine the gradient of $F _ { n }$ and the gradient of $h _ { n }$ at every point of $U _ { n }$.

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $H _ { n }$ the set $H _ { n } = \left\{ \left( x _ { 1 } , \ldots , x _ { n } \right) \in \mathbb { R } ^ { n } \mid x _ { 1 } + \cdots + x _ { n } = 1 \right\}$.
Prove that the restriction of $F _ { n }$ to $\overline { U _ { n } } \cap H _ { n }$ admits a maximum.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Let $V$ be a convex open set of $E$ and $h$ a function of class $\mathscr{C}^1$ from $V$ to $E$. Suppose that there exists a real number $C \geqslant 0$ such that for all $x \in V$, $\|dh(x)\| \leqslant C$. Show that for all $x_1$ and $x_2$ in $V$, we have $\left\|h(x_2) - h(x_1)\right\| \leqslant C \left\|x_2 - x_1\right\|$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Show that there exists a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$

We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
For every real number $r > 0$, we denote by $F(r) = J(p(rX))$.
Justify that the map $F : \mathbf{R}_+^* \rightarrow S_n(\mathbf{R})$ is differentiable and that $$F'(1) = 2n(p(S))^\top p(S) - 2S^\top (p'(S))^\top p(S) - 2(p(S))^\top p'(S) S.$$