Let $(E_n)_{n\in\mathbb{N}}$ be a sequence of finite subsets of $[-1,1]^2$ such that, for all $(u,v)\neq(0,0)$, $$\frac{1}{|E_n|}\sum_{(s,t)\in E_n} e_{u,v}(s,t) \underset{n\rightarrow+\infty}{\longrightarrow} 0.$$ Show that for all $f \in \mathcal{T}$, $$\frac{1}{|E_n|}\sum_{(s,t)\in E_n} f(s,t) \underset{n\rightarrow+\infty}{\longrightarrow} \frac{1}{4}\int_{-1}^{1}\int_{-1}^{1} f(s,t)\,\mathrm{d}s\,\mathrm{d}t.$$

Let $(E_n)_{n\in\mathbb{N}}$ be a sequence of finite subsets of $[-1,1]^2$ such that, for all $(u,v)\neq(0,0)$, $$\frac{1}{|E_n|}\sum_{(s,t)\in E_n} e_{u,v}(s,t) \underset{n\rightarrow+\infty}{\longrightarrow} 0.$$ Show that for all $a,b,c,d \in [-1,1]$ such that $a < b$ and $c < d$, $$\frac{|E_n \cap ([a,b]\times[c,d])|}{|E_n|} \underset{n\rightarrow+\infty}{\longrightarrow} \frac{|b-a||d-c|}{4}.$$

Let $z \in D$. Show that the function $\Phi : t \mapsto L(tz)$ is differentiable on an open interval including $[-1,1]$ and give a simple expression for its derivative on $[-1,1]$.

We are given two matrices $A$ and $B$ in $\mathcal{M}_n(\mathbf{K})$. We assume that $A$ and $B$ commute.
We define an application $$\begin{aligned} g : \mathbf{R} & \rightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto g(t) = e^{t(A+B)} e^{-tB} \end{aligned}$$
$\mathbf{2}$ ▷ Show that the application $g$, and the application $f_A$ defined in the preamble, are solutions to the same Cauchy problem. Deduce from this a proof of the relation $$\forall t \in \mathbf{R} \quad e^{t(A+B)} = e^{tA} e^{tB}.$$

Let $B$ be the matrix in $M_{2}(\mathbf{R})$ defined by: $$B = \left(\begin{array}{cc} 3 & 2 \\ -5 & 1 \end{array}\right)$$ Prove that $B$ is semi-simple and deduce the existence of an invertible matrix $Q$ in $M_{2}(\mathbf{R})$ and two real numbers $a$ and $b$ to be determined such that: $$B = Q \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right) Q^{-1}$$ Hint: for an eigenvector $V$ of $B$, one may introduce the vectors $W_{1} = \operatorname{Re}(V)$ and $W_{2} = \operatorname{Im}(V)$.

Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1).
Show that if $\operatorname{dim}(V \cap V^{\prime}) \geqslant 1$, then $u_k = u_k^{\prime}$ for all $1 \leqslant k \leqslant \operatorname{dim}(V \cap V^{\prime})$.

If $f$ and $g$ are two power series such that $f \prec g$, show that $\rho(f) \geqslant \rho(g)$.

Let $A \in \mathrm { M } _ { 2 } ( \mathrm { C } )$ be a matrix and let $\eta$ be a strictly positive real number.
(a) For $x \in \mathrm { C } ^ { 2 }$, show that the series $$\sum _ { n } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ is convergent. We denote $$N ( x ) = \sum _ { n = 0 } ^ { \infty } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ the sum of this series.
(b) Show that $x \mapsto N ( x )$ is a norm on $\mathbb { C } ^ { 2 }$, which satisfies the following inequality $$\forall x \in \mathbb { C } ^ { 2 } , \quad N ( A x ) \leqslant ( \rho ( A ) + \eta ) N ( x )$$ (c) Show that there exists a real $C > 0$ such that for every $x \in \mathbb { C } ^ { 2 }$ we have $$\| x \| \leqslant N ( x ) \leqslant C \| x \|$$

(a) If $B \in \mathbf { M } _ { \ell } ( \mathbb { C } )$ is diagonalizable, show that there exists a norm $\| \cdot \| _ { B }$ on $\mathbb { C } ^ { \ell }$ such that $\| B x \| _ { B } \leqslant \rho ( B ) \| x \| _ { B }$ for every $x \in \mathbb { C } ^ { \ell }$. Hint: one may verify that if $P \in \mathbf { G L } _ { \ell } ( \mathbb { C } )$, then $x \mapsto \| P x \|$ is a norm on $\mathbb { C } ^ { \ell }$.
(b) Show that there exists a matrix $C \in \mathbf { M } _ { 2 } ( \mathbb { C } )$ such that, for every norm $N$ on $\mathbb { C } ^ { 2 }$ there exists $y \in \mathbb { C } ^ { 2 }$ such that $N ( C y ) > \rho ( C ) N ( y )$.

Let $\phi : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a map and let $x ^ { * }$ be a fixed point of $\phi$. Let $A \in \mathbf { M } _ { 2 } ( \mathbb { R } )$ be a matrix satisfying $\rho ( A ) < 1$, and let $M > 0$ be a real number. We assume that $\phi$ satisfies $$\forall x \in \mathbb { R } ^ { 2 } , \quad \left\| \phi ( x ) - \phi \left( x ^ { * } \right) - A \left( x - x ^ { * } \right) \right\| \leqslant M \left\| x - x ^ { * } \right\| ^ { 2 } .$$ Show that there exists $\varepsilon > 0$ such that for every $x _ { 0 } \in \mathbb { R } ^ { 2 }$ satisfying $\left\| x _ { 0 } - x ^ { * } \right\| < \varepsilon$, the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ defined by $x _ { n + 1 } = \phi \left( x _ { n } \right)$ (for $n \geqslant 0$ ) converges to $x ^ { * }$ when $n \rightarrow + \infty$.

For a triangular matrix $T = \left( \begin{array} { l l } \lambda & a \\ 0 & \mu \end{array} \right) \in \mathbf { M } _ { 2 } ( \mathbb { C } )$, explicitly compute the successive powers $T ^ { n }$ for $n$ a strictly positive integer.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$. Let $y \in \mathbb{R}^d$, show that $$y = \operatorname{proj}_C(x) \Longleftrightarrow y \in C \text{ and } (x - y) \cdot (z - y) \leqslant 0, \forall z \in C.$$

Let $C \in ]0,1[$. Using a simple example of a function $f$, show that the interpolation inequality $$\forall f \in \mathcal{C}^{1}([0,1]), \quad \|f\|_{\infty} \leqslant \left\|f^{\prime}\right\|_{\infty} + C\left|f\left(x_{1}\right)\right|$$ is false.

The bilinear form $B$ is defined by $$B:\left(\begin{pmatrix}x\\y\\z\end{pmatrix},\begin{pmatrix}x'\\y'\\z'\end{pmatrix}\right)\mapsto 3xx'+3yy'-zz'.$$ Given a vector $v\in V$, the pseudo-orthogonal of $v$ is $v^\perp = \{w\in V \mid B(v,w)=0\}$.
Let $v$ be a non-zero vector of $V$. Show that $v^\perp$ is a vector subspace of $V$ of codimension 1, and that $v^\perp$ is a complement of the line generated by $v$ if and only if $B(v,v)\neq 0$.

Let $a , b , c , d$ be four real numbers such that $a \leqslant b$ and $c \leqslant d$. Let $U$ be an open set of $\mathbb { R } ^ { 2 }$ containing $[ a , b ] \times [ c , d ]$. Let $h : U \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 2 }$.
(a) Show the identity $$h ( b , d ) - h ( a , d ) - h ( b , c ) + h ( a , c ) = \int _ { a } ^ { b } \hat { h } \left( s _ { 1 } \right) d s _ { 1 }$$ where $\hat { h }$ is defined by $$\hat { h } \left( s _ { 1 } \right) = \int _ { c } ^ { d } \frac { \partial ^ { 2 } h } { \partial s _ { 1 } \partial s _ { 2 } } \left( s _ { 1 } , s _ { 2 } \right) d s _ { 2 }$$ (b) Deduce that there exists a point $\left( \bar { s } _ { 1 } , \bar { s } _ { 2 } \right)$ of $[ a , b ] \times [ c , d ]$ such that we have the two equalities $$h ( b , d ) - h ( a , d ) - h ( b , c ) + h ( a , c ) = ( b - a ) \hat { h } \left( \bar { s } _ { 1 } \right) = ( b - a ) ( d - c ) \frac { \partial ^ { 2 } h } { \partial s _ { 1 } \partial s _ { 2 } } \left( \bar { s } _ { 1 } , \bar { s } _ { 2 } \right)$$

Let $I$ be an open interval of $\mathbb { R }$. We are given a function $f : I \rightarrow \mathbb { R }$ of class $\mathcal { C } ^ { 3 }$, such that $f ^ { \prime } ( x ) > 0$ for every $x \in I$. Show that $f$ is bijective from $I$ onto the open interval $f ( I )$. We denote by $g : f ( I ) \rightarrow I$ its inverse function. Recall the value of $g ^ { \prime } ( f ( x ) )$. Express $g ^ { \prime \prime } ( f ( x ) )$ as a function of the successive derivatives of $f$ at $x$.

We keep, until the end of this third part, the hypotheses and notation of question 3.2. For $x , y \in I$ such that $y \neq x$, we set $$H _ { f } ( x , y ) = \frac { x f ( y ) - y f ( x ) } { f ( y ) - f ( x ) }$$ (a) Show that for all $x , y \in I$ such that $y \neq x$ we have $$H _ { f } ( x , y ) = x - f ( x ) \int _ { 0 } ^ { 1 } g ^ { \prime } ( \lambda f ( x ) + ( 1 - \lambda ) f ( y ) ) d \lambda$$ (b) Deduce that $H _ { f }$ admits a unique continuous extension to $I \times I$ as a whole. We still denote this extension by $H _ { f } : I \times I \rightarrow \mathbb { R }$.
(c) Show that $H _ { f }$ is of class $\mathcal { C } ^ { 2 }$ on $I \times I$.
(d) Compute $H _ { f } ( x , x )$.

We keep the hypotheses and notation of question 3.2. We now assume $0 \in f ( I )$ and we denote $x ^ { * } = g ( 0 )$. For $x \in I$ we denote by $I _ { x }$ the closed interval with endpoints $x$ and $x ^ { * }$.
(a) Let $x , y \in I$. Show that there exists $( \bar { x } , \bar { y } ) \in I _ { x } \times I _ { y }$, such that $$H _ { f } ( x , y ) - x ^ { * } = \left( x - x ^ { * } \right) \left( y - x ^ { * } \right) \frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } ( \bar { x } , \bar { y } )$$ (b) Compute $$\frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } \left( x ^ { * } , x ^ { * } \right)$$ as a function of the derivatives of $f$.

Let $a , b , c , d$ be four real numbers such that $a \leqslant b$ and $c \leqslant d$. Let $U$ be an open set of $\mathbb { R } ^ { 2 }$ containing $[ a , b ] \times [ c , d ]$. Let $h : U \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 2 }$.
(a) Show the identity $$h ( b , d ) - h ( a , d ) - h ( b , c ) + h ( a , c ) = \int _ { a } ^ { b } \hat { h } \left( s _ { 1 } \right) d s _ { 1 }$$ where $\hat { h }$ is defined by $$\hat { h } \left( s _ { 1 } \right) = \int _ { c } ^ { d } \frac { \partial ^ { 2 } h } { \partial s _ { 1 } \partial s _ { 2 } } \left( s _ { 1 } , s _ { 2 } \right) d s _ { 2 }$$
(b) Deduce that there exists a point $\left( \bar { s } _ { 1 } , \bar { s } _ { 2 } \right)$ of $[ a , b ] \times [ c , d ]$ such that we have the two equalities $$h ( b , d ) - h ( a , d ) - h ( b , c ) + h ( a , c ) = ( b - a ) \hat { h } \left( \bar { s } _ { 1 } \right) = ( b - a ) ( d - c ) \frac { \partial ^ { 2 } h } { \partial s _ { 1 } \partial s _ { 2 } } \left( \bar { s } _ { 1 } , \bar { s } _ { 2 } \right)$$

We keep, until the end of this third part, the hypotheses and notation of the previous question. For $x , y \in I$ such that $y \neq x$, we set $$H _ { f } ( x , y ) = \frac { x f ( y ) - y f ( x ) } { f ( y ) - f ( x ) }$$
(a) Show that for all $x , y \in I$ such that $y \neq x$ we have $$H _ { f } ( x , y ) = x - f ( x ) \int _ { 0 } ^ { 1 } g ^ { \prime } ( \lambda f ( x ) + ( 1 - \lambda ) f ( y ) ) d \lambda$$
(b) Deduce that $H _ { f }$ admits a unique continuous extension to $I \times I$ as a whole. We still denote this extension by $H _ { f } : I \times I \rightarrow \mathbb { R }$.
(c) Show that $H _ { f }$ is of class $\mathcal { C } ^ { 2 }$ on $I \times I$.
(d) Compute $H _ { f } ( x , x )$.

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. For all $x \in [0,1]$ and $f \in \mathcal{C}^{2}([0,1])$, prove the inequality $$\left|f^{\prime}(x) - \frac{f\left(x_{2}\right) - f\left(x_{1}\right)}{x_{2} - x_{1}}\right| \leqslant \left\|f^{\prime\prime}\right\|_{\infty}.$$

Illustrate the construction of the secant method by means of a figure. When $f ^ { \prime } > 0$ on $I$, express $x _ { n + 1 }$ as a function of $x _ { n - 1 } , x _ { n }$ by means of the function $H _ { f }$ defined in question 3.3 of the third part.
Recall: The secant method is defined as follows. Given $x_0, x_1 \in I$, for $n \geq 1$ we consider the line $L_n$ passing through the points $(x_{n-1}, f(x_{n-1}))$ and $(x_n, f(x_n))$ (with $L_n$ being the tangent line at $(x_n, f(x_n))$ when $x_n = x_{n-1}$). If $L_n$ intersects $\{(x,0) \mid x \in I\}$ at a unique point $(x,0)$, we define $x_{n+1} = x$.

In this question, we examine the special case of a polynomial function of degree two $f$ defined by the formula $f ( x ) = ( x - \alpha ) ( x - \beta )$ where $\alpha$ and $\beta$ are real and $\alpha > \beta$. We take $I = ] ( \alpha + \beta ) / 2 , + \infty [$. For $x \in \mathbb { R }$ we define $h ( x ) = \frac { x - \alpha } { x - \beta }$, with the convention $h ( \beta ) = \infty$.
(a) For $x \in \mathbb { R }$ show that we have $| h ( x ) | < 1$ if and only if $x \in I$.
(b) Explicitly state the recurrence relation satisfied by the sequence $u _ { n } : = h \left( x _ { n } \right)$ and deduce that the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined for any $x _ { 0 }$ and $x _ { 1 }$ in $I$.
(c) Show that the sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ tends to 0 and deduce that $\left( x _ { n } \right) _ { n \geqslant 0 }$ tends to $\alpha$.
(d) Let $\phi = \frac { 1 + \sqrt { 5 } } { 2 }$. Show that there exists a strictly negative real number $s$ such that $$x _ { n } - \alpha = \mathrm { O } \left( e ^ { s \phi ^ { n } } \right)$$

We return to the general case, $f$ being any function of class $\mathcal { C } ^ { 3 }$. We assume that $f$ vanishes at a point $x ^ { * } \in I$, for which $f ^ { \prime } \left( x ^ { * } \right) > 0$.
(a) Show that there exists $\epsilon > 0$ such that $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] \subset I$ and $f ^ { \prime } > 0$ on the interval $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. We fix such an $\epsilon$ for the rest and we define $$M = \sup _ { ( x , y ) \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] ^ { 2 } } \left| \frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } ( x , y ) \right|$$ (b) We assume that $x _ { n - 1 } , x _ { n } \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. Show that $$\left| x _ { n + 1 } - x ^ { * } \right| \leqslant M \left| x _ { n - 1 } - x ^ { * } \right| \cdot \left| x _ { n } - x ^ { * } \right| .$$ (c) We fix $\left. \epsilon ^ { \prime } \in \right] 0 , \epsilon$ ] such that $M \epsilon ^ { \prime } < 1$. Show that if $x _ { 0 } , x _ { 1 }$ belong to $\left[ x ^ { * } - \epsilon ^ { \prime } , x ^ { * } + \epsilon ^ { \prime } \right]$ then the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined and converges to $x ^ { * }$.

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Deduce that, for any function $f \in \mathcal{C}^{2}([0,1])$, we have $$\left\|f^{\prime}\right\|_{\infty} \leqslant \left\|f^{\prime\prime}\right\|_{\infty} + \frac{\left|f\left(x_{1}\right)\right| + \left|f\left(x_{2}\right)\right|}{x_{2} - x_{1}}.$$