Specify the term of a function $j$ defined on $\mathbb { R }$ and invertible that satisfies the following condition: The graph of $j$ and the graph of the inverse function of $j$ have no common point.
Given is the function $f : x \mapsto 2 - \ln ( x - 1 )$ with maximal domain $D _ { f }$. The graph of $f$ is denoted by $G _ { f }$.
(1a) [3 marks] Show that $\left. D _ { f } = \right] 1 ; + \infty [$ and specify the behavior of $f$ at the boundaries of the domain.
(1b) [2 marks] Calculate the zero of $f$.
(1c) [5 marks] Describe how $G _ { f }$ is obtained step by step from the graph of the function $x \mapsto \ln x$ defined in $\mathbb { R } ^ { + }$. Use this to explain the monotonicity behavior of $G _ { f }$.
(1d) [4 marks] Show that $F : x \mapsto 3 x - ( x - 1 ) \cdot \ln ( x - 1 )$ with domain $\left. D _ { F } = \right] 1 ; + \infty [$ is an antiderivative of $f$, and determine the term of the antiderivative of $f$ that has a zero at $x = 2$.
Figure 1 shows an obstacle element in a skate park.
[Figure]
Fig. 1
The ramp of the symmetric obstacle element transitions into a horizontally running plateau, which is followed by the descent. The front and rear side surfaces run perpendicular to the horizontal ground. To describe the front side surface mathematically, a Cartesian coordinate system is chosen such that the x-axis represents the lower boundary and the y-axis represents the axis of symmetry of the surface in question. In the model, the plateau extends in the range $- 2 \leq x \leq 2$. The profile line of the descent is described for $2 \leq x \leq 8$ by the graph of the function $f$ investigated in Task 1 (see Figure 2). Here, one unit of length in the coordinate system corresponds to one meter in reality. [Figure]
(2a) [2 marks] Explain the meaning of the function value $f ( 2 )$ in the context of the problem and specify the term of the function $q$ whose graph $G _ { q }$ describes the profile line of the ramp in the model for $- 8 \leq x \leq - 2$.
(2b) [5 marks] Calculate the point $x _ { m }$ in the interval [ $2 ; 8$ ] where the local rate of change of $f$ equals the average rate of change over this interval.
(2c) [3 marks] The value $x _ { m }$ determined by calculation in Task 2b could alternatively be determined approximately without calculation using Figure 2. Explain how you would proceed.
(2d) [2 marks] Based on the model, calculate the size of the angle $\alpha$ that the plateau and the roadway enclose at the edge to the descent (see Figure 2).
(2e) [3 marks] The front side surface of the obstacle element is used as advertising space in partial areas of the ramp and descent (see Figure 1). In the model, these are two surface pieces, namely the area between $G _ { f }$ and the x-axis in the range $2 \leq x \leq 6$ and the corresponding symmetric area in the second quadrant. Using the antiderivative $F$ specified in Task 1d, calculate how many square meters are available as advertising space.
Consider the family of functions $g _ { k } : x \mapsto k x ^ { 3 } + 3 \cdot ( k + 1 ) x ^ { 2 } + 9 x$ defined on $\mathbb { R }$ with $k \in \mathbb { R } \backslash \{ 0 \}$ and the corresponding graphs $G _ { k }$. For each $k$, the graph $G _ { k }$ has exactly one inflection point $W _ { k }$.
(3a) [2 marks] Specify the behavior of $g _ { k }$ at the boundaries of the domain in dependence on $k$.
(3b) [3 marks] Determine the x-coordinate of $W _ { k }$ in dependence on $k$. (for verification: $x = - \frac { 1 } { k } - 1$ )
(3c) [4 marks] Determine the value of $k$ such that the corresponding inflection point $W _ { k }$ lies on the y-axis. Show that in this case the point $W _ { k }$ lies at the origin and the inflection tangent, i.e., the tangent to $G _ { k }$ at the point $W _ { k }$, has slope 9.
(3d) [2 marks] For the value of $k$ determined in Task 3c, Figure 3 shows the corresponding graph with its inflection tangent. In this coordinate system, the two axes have different scales. Determine the missing numerical values at the tick marks on the y-axis using an appropriate slope triangle on the inflection tangent and enter the numerical values in Figure 3. [Figure]

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
Show that if $f$ is a solution of $(E)$ on an interval $J$, and if $a$ is a nonzero real number, then the function $h$ defined by $h(x) = a f\left(\frac{x}{a}\right)$ is also a solution of $(E)$ on an interval that one will specify.

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
We denote $\mathscr{C}$ the image in $\mathbb{R}^2$ of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
We denote $g$ the function of a real variable with real values whose graph is $\gamma\left(\left[\frac{\pi}{4}, \pi\right]\right)$.
II.B.1) Determine the domain of definition $\Delta$ of $g$, as well as an expression for $g$. II.B.2) Verify that the restriction of $g$ to the largest open interval contained in $\Delta$ is a solution of $(E)$. II.B.3) Is this a maximal solution? If not, determine a maximal solution $m$ whose graph includes that of $g$.

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
II.C.1) Recall the statement of the existence and uniqueness theorem for maximal solutions of a nonlinear scalar differential equation subject to Cauchy conditions. II.C.2) Explain how, and possibly to what extent, this theorem applies to $(E)$. II.C.3) Are the maximal solutions given by this theorem maximal solutions of $(E)$? II.C.4) Deduce from the previous questions the maximal solutions of $(E)$.

For $n \in \mathbb { Z }$, we denote by $\mathcal { E } _ { n }$ the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$$
Determine $\mathcal { E } _ { n }$ for $n \in \mathbb { Z }$. We will discuss separately the case $n = 0$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$
Show that $c _ { n , f }$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad \left( c _ { n , f } \right) ^ { \prime } ( r ) = \frac { i n } { r } c _ { n , g } ( r )$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$ For $n \in \mathbb { Z }$, $\mathcal { E } _ { n }$ denotes the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$ for all $t \in \mathbb { R } _ { + } ^ { * }$.
Show that $c _ { n , f }$ belongs to $\mathcal { E } _ { n }$ and that $c _ { n , f }$ is bounded in a neighbourhood of 0. Deduce the existence of $a _ { n } \in \mathbb { C }$ such that $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = a _ { n } r ^ { | n | }$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$. We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
If $n \in \mathbb { Z }$, show that the function $\left( c _ { n , f } \right) ^ { \prime }$ is bounded on $\mathbb { R } _ { + } ^ { * }$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
Show that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are constant.

We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$, that is, the maps from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ of the form $$( x , y ) \mapsto a x ^ { 2 } + b x y + c y ^ { 2 } + d x + e y + f \quad \text { where } \quad ( a , b , c , d , e , f ) \in \mathbb { R } ^ { 6 }$$ A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$
Determine the functions in $\mathcal { P } _ { 2 }$ satisfying (1) on $\mathbb { R } ^ { 2 }$.

If $I$ is an interval of $\mathbb { R }$, we say that $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ satisfies (II.1) on $I$ if and only if $$\forall t \in I , \quad u ( t ) \left( u ( t ) + 2 t u ^ { \prime } ( t ) \right) = - 1$$
By stating precisely the theorem used, show that if $( t _ { 0 } , u _ { 0 } )$ is in $\left( \mathbb { R } ^ { * } \right) ^ { 2 }$, there exist an open interval $I$ of $\mathbb { R }$ containing $t _ { 0 }$ and a function $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ such that $u$ is a solution of (II.1) on $I$ and satisfies $u \left( t _ { 0 } \right) = u _ { 0 }$.

If $I$ is an interval of $\mathbb { R }$, we say that $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ satisfies (II.1) on $I$ if and only if $$\forall t \in I , \quad u ( t ) \left( u ( t ) + 2 t u ^ { \prime } ( t ) \right) = - 1$$ A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ Let $J$ be a non-empty open interval of $\mathbb { R }$, $\Omega ( J ) = \left\{ ( x , y ) \in \mathbb { R } ^ { 2 } , x y \in J \right\}$, $w$ in $\mathcal { C } ^ { 2 } ( J , \mathbb { R } )$ and $W$ the function defined by $$\forall ( x , y ) \in \Omega ( J ) , \quad W ( x , y ) = w ( x y )$$

1. [II.D.1)] Show that $\Omega ( J )$ is a non-empty open set.
2. [II.D.2)] Show that $W$ is in $\mathcal { C } ^ { 2 } ( \Omega ( J ) , \mathbb { R } )$ and that there is equivalence between
   1. [i.] $W$ satisfies (1) on $\Omega ( J )$,
   2. [ii.] $w ^ { \prime }$ satisfies (II.1) on $J$.
3. [II.D.3)] Show that $W$ is the restriction to $\Omega ( J )$ of a function in $\mathcal { P } _ { 2 }$ if and only if $w$ is affine.

A function $f \in \mathcal{C}^2(\Omega, \mathbb{R})$ satisfies (1) on $\Omega$ if and only if $$\forall ( x , y ) \in \Omega, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ Let $\Omega$ be a non-empty open set of $\mathbb { R } ^ { 2 }$, $f$ in $\mathcal { C } ^ { 2 } ( \Omega , \mathbb { R } )$ satisfying (1) on $\Omega$, $( a , b ) \in \mathbb { R } ^ { 2 }$, $\Omega _ { a , b }$ the image of $\Omega$ by the translation of vector $( a , b )$ and $f _ { a , b }$ the function defined on $\Omega _ { a , b }$ by $$\forall ( x , y ) \in \Omega _ { a , b } , \quad f _ { a , b } ( x , y ) = f ( x - a , y - b )$$
Show that $f _ { a , b }$ satisfies (1) on $\Omega _ { a , b }$.

A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) on $\mathbb{R}^2$ if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$.
If $( x _ { 0 } , y _ { 0 } )$ is in $\mathbb { R } ^ { 2 }$, show that there exists an open set $U$ of $\mathbb { R } ^ { 2 }$ containing $( x _ { 0 } , y _ { 0 } )$ such that the set of functions in $\mathcal { C } ^ { 2 } ( U , \mathbb { R } )$ satisfying (1) on $U$ and not coinciding on $U$ with any element of $\mathcal { P } _ { 2 }$ is infinite.

Recall the definition of a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$ and the theorem characterizing such a diffeomorphism among applications of class $\mathcal { C } ^ { 1 }$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R } ^ { 2 }$.

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$ Let $p$ and $q$ be in $\mathbb { R } ^ { 2 }$.

1. [III.B.1)] Verify $$F ( q ) - F ( p ) = \int _ { 0 } ^ { 1 } d F _ { p + t ( q - p ) } ( q - p ) \mathrm { d } t$$
2. [III.B.2)] Show $$\langle F ( q ) - F ( p ) , q - p \rangle \geqslant \alpha \| q - p \| ^ { 2 }$$

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$ Let $a \in \mathbb { R } ^ { 2 }$ and $G ^ { a }$ be the map from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ defined by $$\forall p \in \mathbb { R } ^ { 2 } , \quad G ^ { a } ( p ) = \| F ( p ) - a \| ^ { 2 }$$

1. [III.C.1)] If $p$ and $h$ are in $\mathbb { R } ^ { 2 }$, compute $d G ^ { a } { } _ { p } ( h )$.
2. [III.C.2)] Show that $G ^ { a } ( p ) \rightarrow + \infty$ when $\| p \| \rightarrow + \infty$.
3. [III.C.3)] Deduce that $G ^ { a }$ attains a global minimum on $\mathbb { R } ^ { 2 }$ at a point $p _ { 0 }$.
4. [III.C.4)] Show that $F \left( p _ { 0 } \right) = a$.

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$
Show that $F$ realizes a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ For $( x , y ) \in \mathbb { R } ^ { 2 }$, let $u ( x , y ) = x + \frac { \partial f } { \partial x } ( x , y ) , v ( x , y ) = y + \frac { \partial f } { \partial y } ( x , y )$ and $F ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. We assume that $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) > 0$ for all $( x , y ) \in \mathbb { R } ^ { 2 }$.
If $( x , y ) \in \mathbb { R } ^ { 2 }$, show that $\operatorname { Jac } F ( x , y ) - I _ { 2 }$ (where $I _ { 2 }$ denotes the identity matrix of order 2) is symmetric positive semidefinite. Deduce that $F$ is a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ For $( x , y ) \in \mathbb { R } ^ { 2 }$, let $u ( x , y ) = x + \frac { \partial f } { \partial x } ( x , y ) , v ( x , y ) = y + \frac { \partial f } { \partial y } ( x , y )$ and $F ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. We assume that $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) > 0$ for all $( x , y ) \in \mathbb { R } ^ { 2 }$. Let $r ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) , s ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y )$ and $t ( x , y ) = \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y )$ so that, for all $( x , y ) \in \mathbb { R } ^ { 2 }$, $r ( x , y ) > 0$ and $r ( x , y ) t ( x , y ) - s ( x , y ) ^ { 2 } = 1$.

1. [IV.B.1)] Show that there exist two functions $\varphi$ and $\psi$ in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ such that $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \left\{ \begin{array} { l } \varphi ( u ( x , y ) , v ( x , y ) ) = x - \frac { \partial f } { \partial x } ( x , y ) \\ \psi ( u ( x , y ) , v ( x , y ) ) = - y + \frac { \partial f } { \partial y } ( x , y ) \end{array} \right.$$
2. [IV.B.2)] Compute $\frac { \partial \varphi } { \partial u } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \varphi } { \partial v } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \psi } { \partial u } ( u ( x , y ) , v ( x , y ) )$ and $\frac { \partial \psi } { \partial v } ( u ( x , y ) , v ( x , y ) )$ (which we will abbreviate as $\frac { \partial \varphi } { \partial u } , \frac { \partial \varphi } { \partial v } , \frac { \partial \psi } { \partial u }$ and $\frac { \partial \psi } { \partial v }$) in terms of $r ( x , y ) , s ( x , y )$ and $t ( x , y )$ (which we will abbreviate as $r , s$ and $t$).
3. [IV.B.3)] Show that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are bounded on $\mathbb { R } ^ { 2 }$.
4. [IV.B.4)] Show, using the first part, that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are constant.
5. [IV.B.5)] Deduce that $r , s$ and $t$ are constant.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$.
Show that the only functions in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$ belong to $\mathcal { P } _ { 2 }$.

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Show that $f'' = 0$, then that the set of continuous solutions of equation (IV.1) forms an $\mathbb{R}$-vector space, for which we will determine a basis.

Let $T$ be a strictly positive real number. We denote by $E(T)$ the set consisting of pairs $(u,v)$ of continuous functions on $[0,T]$ with real values.
A Carnot path controlled by $(u,v) \in E(T)$ is a map $\gamma : [0,T] \rightarrow \mathcal{M}_3(\mathbf{R})$ of class $C^1$ solution of the matrix differential equation: $$\left\{\begin{array}{l} \gamma'(t) = u(t)\gamma(t)M_{1,0,0} + v(t)\gamma(t)M_{0,1,0} \\ \gamma(0) = I_3 \end{array}\right.$$ where $M_{1,0,0}$ and $M_{0,1,0}$ are as defined in the first part.
(a) For all $(u,v) \in E(T)$, justify the existence of a unique Carnot path controlled by $(u,v)$.
(b) Show that $\gamma$ satisfies $$\forall t \in [0,T], \quad \gamma(t) \in \mathbf{H}$$ and explicitly calculate, as a function of $t$, $u$ and $v$, the functions $p(t)$, $q(t)$ and $r(t)$ such that $$\gamma(t) = \exp\left(M_{p(t),q(t),r(t)}\right).$$

For all $(\theta, \varphi) \in \mathbf{R}^2$ and $t \in \mathbf{R}$, we define the controls $$u_{\theta,\varphi}(t) = \sin(\theta - \varphi t) \quad \text{and} \quad v_{\theta,\varphi}(t) = \cos(\theta - \varphi t)$$ and we denote $\gamma_{\theta,\varphi}(t) = \exp\left(M_{p(t),q(t),r(t)}\right)$ the Carnot path controlled by $(u_{\theta,\varphi}, v_{\theta,\varphi})$.
(a) We assume $\varphi \neq 0$. Calculate $p(t)$ and $q(t)$ and verify that $$r(t) = \frac{t\varphi - \sin(t\varphi)}{2\varphi^2}$$
(b) Similarly calculate $\gamma_{\theta,0}(t)$.

The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
We define the functions $f$ and $g$ on $]0, 2\pi]$ by: $$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$
Show that $f$ and $g$ extend by continuity to $[0, 2\pi]$; that $f$ is then a continuous bijection from $[0, 2\pi]$ onto a set to be specified; and that $g$ attains its maximum at $\pi$.