Study the variations of $g_{\sigma}$. Show that the second derivative of $g_{\sigma}$ vanishes and changes sign at exactly two points. Give the shape of the graph of $g_{\sigma}$ and mark the two points mentioned.

Give the shape of the representative curve of $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. Determine an equivalent of $W ( x )$ as $x \rightarrow 0$ as well as an equivalent of $W ( x )$ as $x \rightarrow + \infty$.

Let $f(x) = xe^x$ and let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$. Sketch, on the same diagram, the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { W }$ representing the functions $f$ and $W$. Specify the tangent lines to the two curves at the point with abscissa 0 as well as the tangent line to $\mathcal { C } _ { W }$ at the point with abscissa $- \mathrm { e } ^ { - 1 }$.

For all $s \in [0,1]$, the function $k_s$ is defined by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Let $s \in ]0,1[$. Sketch the graph of $k_s$ on $[0,1]$.

The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $q$ is piecewise continuous on $\mathbf { R }$, that it is 1-periodic, and that the function $| q |$ is even.

In this part, we introduce the function $q$ which associates to any real $x$ the real number $q(x) = x - \lfloor x \rfloor - \frac{1}{2}$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $q$ is piecewise continuous on $\mathbf{R}$, that it is 1-periodic and that the function $|q|$ is even.

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Sketch the graph of $f$ by making best use of the previous results.

Study the convexity of the function $t \mapsto \ln(1 + \mathrm{e}^t)$.

Study the convexity of the function $t \mapsto \ln \left( 1 + \mathrm { e } ^ { t } \right)$.

We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$ For $n \in \mathbf{N}^*$, set $a_n = \frac{1}{n+1}$ and $b_n = \frac{n}{n+1}$.
Establish the pointwise convergence of the sequence of applications $(\Psi_n)_{n \in \mathbf{N}^*}$, from $]0, \pi]$ to $\mathbf{R}$, defined by: $$\forall n \in \mathbf{N}^*, \forall t \in ]0, \pi], \Psi_n(t) = \ln(a_n^2 \cos^2 t + b_n^2 \sin^2 t)$$ Deduce $f''(0)$.

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0$$ and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Justify the existence of the real number $\Delta _ { n }$ for all $n \in \mathbb { N } ^ { * }$.

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0$$ and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Show that, for all $n \in \mathbb { N } ^ { * }$, we have the equality $$\Delta _ { n } = \sup _ { x \geqslant 0 } \left| B _ { n } ( x ) - \varphi ( x ) \right| .$$

Consider the function $$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$ Justify that $f \in \mathcal{C}^1(\mathbb{R})$ and that $f$ is convex. Give the set of its minimizers.

101- For which set of values of $a$, the graph of $f(x) = (a-3)x^2 + ax - 1$ does not pass through the first quadrant?

• [(1)] $a \leq 2$
• [(2)] $0 < a \leq 2$
• [(3)] $2 < a < 3$
• [(4)] $0 < a < 3$

115- If $f(x) = [x] + [-x]$ and $g(x) = \begin{cases} f(x) & ; \ x \notin \mathbb{Z} \\ f(x) - 1 & ; \ x \in \mathbb{Z} \end{cases}$, then the number of points of discontinuity of $g$ on the interval $[4, -4]$ is which of the following?
(1) $1$ (2) $2$ (3) $3$ (4) zero

122- The figure opposite shows the graph of $f(x) = \dfrac{x^3 + ax^2}{x^2 + bx + c}$. The value of $(bc - a)$ is which of the following?
[Figure: Graph of a rational function with asymptotes, showing a curve with a local feature near the origin]
(1) $-2$ (2) $-1$ (3) $1$ (4) $2$

107- For which values of $x$ in the domain of $y=\sqrt{5+4x-x^2}$, does the graph of $y=\sqrt{5+4x-x^2}$ lie above the graph of $y=|x-3|+2$?
(1) $\left(\dfrac{3-\sqrt{17}}{2},\ 5\right)$ (2) $\left(2,\ \dfrac{3+\sqrt{17}}{2}\right)$
(3) $\left(2,\ \dfrac{4+\sqrt{15}}{2}\right)$ (4) $\left(2,\ 2+\sqrt{15}\right)$

117. The figure on the left shows the graph of the function $y = f(x)$. The graph of $f'(x)$ is in which form?
[Figure: The main graph shows an S-shaped (sigmoid-like) increasing curve with two horizontal asymptotes.]
[Option (1): Graph with a sharp peak (cusp) at the origin, symmetric, going to zero on both sides.]
[Option (2): Graph with a smooth bell-shaped curve (positive hump).]
[Option (3): Graph with a curve that dips below the x-axis on the left and rises above on the right, with horizontal asymptotes.]
[Option (4): Graph with a smooth curve having a negative dip, symmetric about y-axis, with horizontal asymptotes.]

116. The extensions of the asymptotes of the graph of the function $f(x) = \sqrt{x^2 + 2x} - \sqrt{x^2 - 2x}$, the first and third bisectors intersect at two points A and B. What is the length of AB?
(1) $2\sqrt{7}$ (2) $4$ [6pt] (3) $2\sqrt{5}$ (4) $4\sqrt{7}$

109. The graph of $y = \cos(\tan^{-1} x)$ and the line $y = mx$, for which set of values of $m$, share exactly one point in common?
(4) $(0, +\infty)$ (3) $(-\infty, 0)$ (2) $(-\infty, +\infty)$ (1) $(-\infty, +\infty) - \{0\}$

116. The oblique asymptote of the curve $y = \sqrt[2]{8x^2 + 2x^2}$ intersects the $y$-axis at which point?
(4) $\dfrac{5}{6}$ (3) $\dfrac{2}{3}$ (2) $\dfrac{1}{2}$ (1) $\dfrac{1}{6}$

122- The figure opposite shows part of the graph of $f(x) = \dfrac{x^2 + ax + b}{x + c}$. What is $b$?
[Figure: Graph of a rational function with vertical asymptote and oblique asymptote; marked points at $x = -2$ and $x = 7$ on the x-axis]
(1) $1$
(2) $4$
(3) $6$
(4) $9$

115- The $x$-intercept of the oblique asymptote of $y = x\sqrt{\dfrac{3x-3}{x-1}}$ is which of the following?
(1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{4}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{3}{4}$

117. Suppose $f(x) = \begin{cases} (x-1)|x| & ; \ |x-1| < 1 \\ x^2 + ax + b & ; \ |x-1| \geq 1 \end{cases}$ is always a continuous function. What is the value of $a$?
(1) $\dfrac{3}{2}$ (2) $-1$ (3) $1$ (4) $\dfrac{5}{2}$