For a real number $t$, let $f(t)$ denote the slope of the line passing through the origin and tangent to the curve $y = \frac{1}{e^x} + e^t$. For the constant $a$ satisfying $f(a) = -e\sqrt{e}$, find the value of $f'(a)$. [3 points]
(1) $-\frac{1}{3}e\sqrt{e}$
(2) $-\frac{1}{2}e\sqrt{e}$
(3) $-\frac{2}{3}e\sqrt{e}$
(4) $-\frac{5}{6}e\sqrt{e}$
(5) $-e\sqrt{e}$

16. The tangent line to the curve $y = x + \ln x$ at the point $( 1,1 )$ is tangent to the curve $y = a x ^ { 2 } + ( a + 2 ) x + 1$. Then $a = $ $\_\_\_\_$ .

III. Solution Questions

17 (This question is worth 12 points). In $\triangle A B C$, $D$ is a point on $BC$, $AD$ bisects $\angle B A C$, and $B D = 2 D C$.
(1) Find $\frac { \sin \angle B } { \sin \angle C }$; (II) If $\angle B A C = 60 ^ { \circ }$, find $\angle B$.

The slope of the tangent line to the curve $y = ( ax + 1 ) e ^ { x }$ at the point $( 0,1 )$ is $- 2$. Then $a = $ $\_\_\_\_$.

Given the function $f ( x ) = \frac { \mathrm { e } ^ { x } } { x + a }$. If $f ^ { \prime } ( 1 ) = \frac { \mathrm { e } } { 4 }$, then $a =$ $\_\_\_\_$ .

Let $n \in \mathbb{N}$.
a) Show that the function $F_n$ is of class $C^\infty$ on $\mathbb{R}$.
b) For $x \in ]-1,1[$, give a simple expression for $F_n'(x)$. Justify the calculation carefully.

Let $n \in \mathbb{N}^*$.
a) Show that $\arccos(x) \sim \sqrt{2(1-x)}$ as $x \rightarrow 1$.
b) Deduce the calculation of $F_n'(1)$ and $F_n'(-1)$.

Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Show that, for all $n \in \mathbb{N}^*$ and all real $x$, the following relation holds: $$\left(1 - x^2\right) T_n''(x) - x T_n'(x) + n^2 T_n(x) = 0.$$

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Show that for all $n \in \mathbb{N}^*$, $$\sup_{x \in [-1,1]} \left| T_n'(x) \right| = 2^{1-n} n^2$$
Is this supremum attained? If so, specify for which values of $x$.

Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
Show that the applications $u$ and $v$ are of class $C^{1}$ on the interval $]0,1[$ and, for each $t \in ]0,1[$, calculate the derivatives $u'(t)$ and $v'(t)$.

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 two functions on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$ by $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta ) \quad \text { and } \quad \widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$$
Express $\frac { \partial \widetilde { f } } { \partial r } ( r , \theta )$ and $\frac { \partial \widetilde { f } } { \partial \theta } ( r , \theta )$ in terms of $r , \theta , \frac { \partial f } { \partial x } ( r \cos \theta , r \sin \theta )$ and $\frac { \partial f } { \partial y } ( r \cos \theta , r \sin \theta )$.

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 two functions on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$ by $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta ) \quad \text { and } \quad \widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$$
For all $( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R }$, show that $\frac { \partial \widetilde { f } } { \partial r } ( r , \theta ) = \frac { 1 } { r } \times \frac { \partial \widetilde { g } } { \partial \theta } ( r , \theta )$ and $\frac { \partial \widetilde { g } } { \partial r } ( r , \theta ) = - \frac { 1 } { r } \times \frac { \partial \widetilde { f } } { \partial \theta } ( r , \theta )$.

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Deduce that the function $f$ is of class $C^\infty$ on $]-\infty, M[$.

Show that the derivative of every element of $\mathcal{D}$ is an element of $\mathcal{D}$.

For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $f$ be a function belonging to $\mathcal{F}_{sr}$. Show that the function $f * \rho_n$ is of class $\mathcal{C}^{\infty}$.

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $f$ is in $\mathcal{S}$ then $f^{(p)}$ is in $\mathcal{S}$ for every natural number $p$.

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
We consider the application $f_\alpha$ defined on $\mathbb{R}_+^*$ by $f_\alpha(x) = x^\alpha$. Show that $\alpha$ is a natural number if and only if one of the successive derivatives of $f_\alpha$ vanishes at least at one strictly positive real.

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$. The notations are those of question IV.A.4.
What is the limit of the expression $f_\alpha^{(n)}(x + y_n)$ when $x \in \mathbb{N}^*$ tends to $+\infty$?

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
IV.A.1) Show that the function $g$ is of class $C^{1}$ on $]-1,1[\backslash\{0\}$ and continuous on $]-1,1[$.
IV.A.2) Calculate the limit of $g'$ at 0. Deduce that $g$ is of class $C^{1}$ on $]-1,1[$.

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. Let $t \in [-1/2, 1/2]$. We consider the function $G_{t}$ defined on $[-1/2, 1/2]$ by
$$\forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad G_{t}(x) = f'(x+t)\sin(\pi x) - (f(x+t)-f(t))\pi\cos(\pi x)$$
Establish the existence of a real number $D$, independent of $x$ and $t$, such that
$$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad |G_{t}(x)| \leqslant D x^{2}$$

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$. Show that $h$ is of class $C^{\infty}$ on $\mathbb{R}$.

Verify that $\varphi$ is of class $\mathscr{C}^{0}$ on $[0, +\infty[$ and $\mathscr{C}^{\infty}$ on $]0, +\infty[$. Give the limit of the derivative $\varphi'(t)$ of $\varphi$ as $t$ tends to 0 in $]0, +\infty[$.
Where $\varphi$ is defined by $$\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$$

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Calculate $\lim _ { \substack { x \rightarrow 1 \\ x < 1 } } \varphi ^ { \prime } ( x )$ and demonstrate that $\varphi$ is of class $C ^ { 1 }$ on $\mathbb { R }$.

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Show that, for all non-zero natural integer $p$, there exist two polynomials $P _ { p }$ and $Q _ { p }$ with real coefficients such that, for all $x \in ] - \infty , 1 [$, $$\varphi ^ { ( p ) } ( x ) = \frac { P _ { p } ( \sqrt { 1 - x } ) } { Q _ { p } ( \sqrt { 1 - x } ) } \exp \left( \frac { - x } { \sqrt { 1 - x } } \right)$$

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Deduce $\lim _ { \substack { x \rightarrow 1 \\ x < 1 } } \varphi ^ { ( p ) } ( x )$ for $p \in \mathbb { N } ^ { * }$.

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Justify that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R } ^ { + * }$ and demonstrate that, for all $n \in \mathbb { N } ^ { * }$, there exists $P _ { n } \in \mathbb { C } [ X ]$ such that $$\forall x \in ] 0 , + \infty \left[ \quad \theta ^ { ( n ) } ( x ) = \frac { P _ { n } ( \ln x ) } { x ^ { n } } \theta ( x ) \right.$$