From now on, $f$ denotes an infinitely differentiable function from $[ 0,1 ]$ to $\mathbb { R }$. We assume that there exists a unique point $x _ { 0 } \in \left[ 0,1 \left[ \right. \right.$ where $f ^ { \prime }$ vanishes. We also assume that $f ^ { \prime \prime } \left( x _ { 0 } \right) > 0$. We are also given an infinitely differentiable function $g : [ 0,1 ] \rightarrow \mathbb { R }$.
For all $x \in \left[ x _ { 0 } , 1 \right]$, we define $$h ( x ) = \sqrt { \left| f ( x ) - f \left( x _ { 0 } \right) \right| }$$ We admit that the bijection $$h : \left\{ \begin{array} { c c c } { \left[ x _ { 0 } , 1 \right] } & \rightarrow & { [ 0 , h ( 1 ) ] } \\ x & \mapsto & h ( x ) \end{array} \right.$$ admits an inverse map $h ^ { - 1 } : [ 0 , h ( 1 ) ] \rightarrow \left[ x _ { 0 } , 1 \right]$ that is infinitely differentiable.
Assume that $\left. x _ { 0 } \in \right] 0,1 [$. Show that, as $t \rightarrow + \infty$, $$\int _ { 0 } ^ { 1 } g ( x ) \sin ( t f ( x ) ) \mathrm { d } x = g \left( x _ { 0 } \right) \sin \left( t f \left( x _ { 0 } \right) + \frac { \pi } { 4 } \right) \sqrt { \frac { 2 \pi } { t f ^ { \prime \prime } \left( x _ { 0 } \right) } } + O \left( \frac { 1 } { t } \right)$$

We keep the hypotheses and notation of questions 3.2 and 3.3. 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 $f$.

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Justify that $f$ is of class $\mathcal{C}^2$, decreasing and convex on $I$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that, if $\varphi_\alpha''(0) > 0$, then there exists $\eta > 0$ such that for all $t \in ]-\eta, \eta[$, $$\frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM) \geq \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A) - \operatorname{Tr}(A^{-1}M) \operatorname{det}^{-\alpha}(A) t.$$

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 $n$ 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 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 denote by $H _ { n }$ the set $H _ { n } = \left\{ \left( x _ { 1 } , \ldots , x _ { n } \right) \in \mathbb { R } ^ { n } \mid x _ { 1 } + \cdots + x _ { n } = 1 \right\}$. We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$.
Prove that there exists a real number $\lambda > 0$ such that
$$\left\{ \begin{aligned} \gamma _ { 1 } + \frac { \gamma _ { 2 } } { 2 } + \cdots + \frac { \gamma _ { n } } { n } & = \lambda a _ { 1 } \\ \frac { \gamma _ { 2 } } { 2 } + \cdots + \frac { \gamma _ { n } } { n } & = \lambda a _ { 2 } \\ & \vdots \\ \frac { \gamma _ { n } } { n } & = \lambda a _ { n } \\ a _ { 1 } + a _ { 2 } + \cdots + a _ { n } & = 1 \end{aligned} \right.$$

We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. Show that $0 \leq f(x) - f(x_*) \leq |x - x_*||f'(x)|$ for all $x \in \mathbb{R}$.

We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := x_n - \tau f'(x_n)$. Show that for all $n \in \mathbb{N}$, assuming $x_0 \neq x_*$, $$f(x_{n+1}) \leq f(x_n) - \frac{\tau}{2}(2 - \tau L)\frac{\left|f(x_n) - f(x_*)\right|^2}{\left|x_0 - x_*\right|^2}$$ Hint: use question 2.c)

We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := x_n - \tau f'(x_n)$. Establish an upper bound for the sequence with general term $a_n := f(x_n) - f(x_*)$. Conclude that $f(x_n) \rightarrow f(x_*)$ when $n \rightarrow \infty$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
By distinguishing cases according to whether $\beta \leqslant 1$ or $\beta > 1$, show that $G_h$ admits a minimum on $\mathbb{R}_+$ attained at a unique point $u_h$. Show moreover that

• [(a)] if $\beta \leqslant 1$ and $h = 0$, then $u_h = 0$;
• [(b)] if $\beta > 1$, then $u_h > 0$;
• [(c)] for any $\beta \in \mathbb{R}_+^*$,
  • [(i)] $G_h'(u_h) = 0$;
  • [(ii)] $h = \beta G_0'(u_h)$;
  • [(iii)] $G_h''(u_h) > 0$ when $h > 0$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
We now assume that $h > 0$. Justify that $G_h$ attains its minimum on $\mathbb{R}$ at a unique point which is $u_h$.

Show that
$$\forall ( x , y ) \in ] 0,1 \left[ ^ { 2 } , \quad \frac { x ( 1 - x ) y ( 1 - y ) } { 1 - x y } \leqslant \frac { 5 \sqrt { 5 } - 11 } { 2 } \right.$$

113- If $f(x) = \text{Max}\{|2x|, |x+1|\}$, then what is the minimum value of $f(x)$?
(1) $\dfrac{1}{3}$ (2) $\dfrac{2}{3}$ (3) $\dfrac{4}{3}$ (4) $2$

116- What is the minimum value of the function $f(x) = x + \sqrt[3]{x^2 - x^3}$?
(1) $-\dfrac{1}{9}$ (2) $-\dfrac{1}{6}$ (3) $-\dfrac{1}{3}$ (4) zero

120- For which set of values of $a$, the curve $y = x^4 + ax^3 + \dfrac{3}{2}x^2$ is always concave up?
(1) $-1 < a < 1$ (2) $-1 < a < 2$ (3) $-2 < a < 1$ (4) $-2 < a < 2$

121- The set of lengths of inflection points of the curve $y = x|x^2 - 4x|$ is which of the following?
(1) $\left\{\dfrac{4}{3}\right\}$ (2) $\left\{0, \dfrac{4}{3}, 4\right\}$ (3) $\left\{\dfrac{4}{3}, 4\right\}$ (4) $\left\{0, \dfrac{4}{3}\right\}$

121. For which domain, the function $f(x) = x^3 e^{-x}$ is increasing and its graph is concave upward?
(1) $(0, 3-\sqrt{3})$ (2) $(3-\sqrt{3}, 3)$ (3) $(3, 3+\sqrt{3})$ (4) $(3+\sqrt{3}, +\infty)$

118. From the relation $x^2y - y^2 - 2\sqrt{x} + 4 = 0$, the value of $\dfrac{d^2y}{dx^2}$ at the point $(1, 2)$ is which of the following?
(1) $\dfrac{7}{6}$ (2) $\dfrac{8}{6}$ [6pt] (3) $\dfrac{11}{6}$ (4) $\dfrac{13}{6}$

119. If $f(x) = x^3 - x^2 + 2x$, the equation of the line perpendicular to the curve of $f^{-1}$ at the point $x = 2$ is which of the following?
(1) $y + 3x = 7$ (2) $y - 3x = -5$ [6pt] (3) $3y + x = 5$ (4) $3y - x = 1$
120. For the graph of $y = |x| \cdot e^{-x}$, on which interval is it decreasing and concave down?
(1) $(-\infty, 2)$ (2) $(0, 1)$ [6pt] (3) $(1, 2)$ (4) $(2, +\infty)$

122- The figure below shows the graph of the function with equation $f(x) = -x^4 + 4x^3 + ax^2 + b$. What is $a$?
[Figure: Graph of a polynomial function with a local maximum and minimum]

• [(1)] $-18$
• [(2)] $-15$
• [(3)] $-12$
• [(4)] $-9$

120- What is the length of the inflection point of the graph of $y = (5-x)\sqrt[3]{x^2}$?
(1) $-1$ (2) zero (3) $1$ (4) $2$

122. For the graph of $f(x) = \cos^2 x - 2\cos x$; $x \in [0, 2\pi]$, at which base point is the inflection point and local minimum?
(1) $\left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ (2) $\left(\pi, \dfrac{4\pi}{3}\right)$ (3) $\left(\dfrac{2\pi}{3}, \pi\right)$ (4) $\left(\dfrac{4\pi}{3}, \dfrac{3\pi}{2}\right)$

122- The figure below shows the graph of $y = \dfrac{x^2 + ax^2}{x^2 + bx + 1}$. What is the value of the relative minimum of the function?
[Figure: Graph of the function with a local minimum visible, axes labeled $x$ and $y$]
(1) $4/5$
(2) $6$
(3) $6/25$
(4) $6/75$

118. The graph shown is the graph of the function $f(x) = 3x^4 + ax^3 + bx^2 + cx$. What is $a$?
[Figure: Graph of a function with a local minimum near $x=1$ on the positive x-axis, with the curve going to positive infinity on both sides]
(1) $-8$ (2) $-7$ (3) $-5$ (4) $-4$