Let $f$ be a function defined and differentiable on $\mathbb{R}$. We denote by $\mathscr{C}$ its representative curve in the plane equipped with a coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$.
Part A
In the graphs below, we have represented the curve $\mathscr{C}$ and three other curves $\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3$ with the tangent line at their point with abscissa 0.

1. Determine by reading the graph, the sign of $f(x)$ according to the values of $x$.
2. We denote by $F$ a primitive of the function $f$ on $\mathbb{R}$. a. Using the curve $\mathscr{C}$, determine $F'(0)$ and $F'(-2)$. b. One of the curves $\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3$ is the representative curve of the function $F$. Determine which one by justifying the elimination of the other two.

Part B
In this part, we assume that the function $f$ mentioned in Part A is the function defined on $\mathbb{R}$ by $$f(x) = (x+2)\mathrm{e}^{\frac{1}{2}x}$$

1. Observation of the curve $\mathscr{C}$ allows us to conjecture that the function $f$ admits a minimum. a. Prove that for all real $x$, $f'(x) = \frac{1}{2}(x+4)\mathrm{e}^{\frac{1}{2}x}$. b. Deduce a validation of the previous conjecture.
2. Let $I = \int_0^1 f(x)\,\mathrm{d}x$. a. Give a geometric interpretation of the real number $I$. b. Let $u$ and $v$ be the functions defined on $\mathbb{R}$ by $u(x) = x$ and $v(x) = \mathrm{e}^{\frac{1}{2}x}$. Verify that $f = 2(u'v + uv')$. c. Deduce the exact value of the integral $I$.
3. The algorithm below is given.
   

   Variables :	$k$ and $n$ are natural integers. $s$ is a real number.
   Input :	Assign to $s$ the value 0.
   Processing :	For $k$ ranging from 0 to $n-1$
   	End of loop.
   Output :	Display $s$.

   
   We denote by $s_n$ the number displayed by this algorithm when the user enters a strictly positive natural integer as the value of $n$. a. Justify that $s_3$ represents the area, expressed in square units, of the shaded region in the graph below where the three rectangles have the same width. b. What can be said about the value of $s_n$ provided by the proposed algorithm when $n$ becomes large?

Exercise 1 -- Common to all candidates

Part A

Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \frac{3}{1 + \mathrm{e}^{-2x}}$$ In the graph below, we have drawn, in an orthogonal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, the representative curve $\mathscr{C}$ of the function $f$ and the line $\Delta$ with equation $y = 3$.

1. Prove that the function $f$ is strictly increasing on $\mathbb{R}$.
2. Justify that the line $\Delta$ is an asymptote to the curve $\mathscr{C}$.
3. Prove that the equation $f(x) = 2.999$ has a unique solution $\alpha$ on $\mathbb{R}$.

Determine an interval containing $\alpha$ with amplitude $10^{-2}$.

Part B

Let $h$ be the function defined on $\mathbb{R}$ by $h(x) = 3 - f(x)$.

1. Justify that the function $h$ is positive on $\mathbb{R}$.
2. We denote by $H$ the function defined on $\mathbb{R}$ by $H(x) = -\frac{3}{2}\ln\left(1 + \mathrm{e}^{-2x}\right)$.
   Prove that $H$ is an antiderivative of $h$ on $\mathbb{R}$.
3. Let $a$ be a strictly positive real number. a. Give a graphical interpretation of the integral $\int_{0}^{a} h(x)\,\mathrm{d}x$. b. Prove that $\int_{0}^{a} h(x)\,\mathrm{d}x = \frac{3}{2}\ln\left(\frac{2}{1 + \mathrm{e}^{-2a}}\right)$. c. We denote by $\mathscr{D}$ the set of points $M(x\,;\,y)$ in the plane defined by $$\left\{\begin{array}{l} x \geqslant 0 \\ f(x) \leqslant y \leqslant 3 \end{array}\right.$$ Determine the area, in square units, of the region $\mathscr{D}$.

The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ Let $V_m$ denote the average rate (as a percentage) of $\mathrm{CO}_2$ present in the room during the first 11 minutes of operation of the extractor hood. a. Let $F$ be the function defined on the interval $[0;11]$ by: $$F(t) = (-1{,}6t - 3{,}6)\mathrm{e}^{-0{,}5t} + 0{,}03t.$$ Show that the function $F$ is an antiderivative of the function $f$ on the interval $[0;11]$. b. Deduce the average rate $V_m$, the average value of the function $f$ on the interval $[0;11]$. Round the result to the nearest thousandth, that is to $0.1\%$.

Consider a function $h$ defined and continuous on $\mathbb{R}$ whose variation table is given below:

$x$	$-\infty$	1	$+\infty$
Variations of $h$		0
	$-\infty$

We denote $H$ the antiderivative of $h$ defined on $\mathbb{R}$ which vanishes at 0. It satisfies the property: a. $H$ is positive on $]-\infty; 0]$. b. $H$ is increasing on $]-\infty; 1]$. c. $H$ is negative on $]-\infty; 1]$. d. $H$ is increasing on $\mathbb{R}$.

The following shows the graph of a continuous function $y = f ( x )$ and two distinct points $\mathrm { P } ( a , f ( a ) ) , \mathrm { Q } ( b , f ( b ) )$ on this graph.
When the function $F ( x )$ satisfies $F ^ { \prime } ( x ) = f ( x )$, which of the following statements in $\langle$Remarks$\rangle$ are always correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. The function $F ( x )$ is increasing on the interval $[ a , b ]$. ㄴ. $\frac { F ( b ) - F ( a ) } { b - a }$ is equal to the slope of line PQ. ㄷ. $\int _ { a } ^ { b } \{ f ( x ) - f ( b ) \} d x \leqq \frac { ( b - a ) \{ f ( a ) - f ( b ) \} } { 2 }$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the function $f ( x ) = 4 x ^ { 3 } - 2 x$, let $F ( x )$ be an antiderivative with $F ( 0 ) = 4$. Find the value of $F ( 2 )$. [3 points]

We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $$F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$$ where $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Verify that $F$ is an antiderivative of $x \mapsto f ( x ) \frac { \mathrm { e } ^ { - x } } { x }$ on the interval $\mathbb { R } _ { + } ^ { * }$.

$\int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}} dx$ is equal to
(1) $x + 2\sin x + \sin 2x + c$
(2) $2x + \sin x + \sin 2x + c$
(3) $x + 2\sin x + 2\sin 2x + c$
(4) $2x + \sin x + 2\sin 2x + c$

If $f ^ { \prime } ( x ) = \tan ^ { - 1 } ( \sec x + \tan x ) , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ and $f ( 0 ) = 0$, then $f ( 1 )$ is equal to:
(1) $\frac { \pi + 1 } { 4 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { \pi - 1 } { 4 }$
(4) $\frac { \pi + 2 } { 4 }$

Let the curve $y = y ( x )$ be the solution of the differential equation, $\frac { d y } { d x } = 2 ( x + 1 )$. If the numerical value of area bounded by the curve $y = y ( x )$ and $x$-axis is $\frac { 4 \sqrt { 8 } } { 3 }$, then the value of $y ( 1 )$ is equal to $\_\_\_\_$.

Let $F(x)$ and $f(x)$ both be polynomial functions with real coefficients. Given that $F'(x) = f(x)$, select the correct options.
(1) If $a \geq 0$, then $F(a) - F(0) = \int_{0}^{a} f(t)\, dt$
(2) If $F(x)$ divided by $x$ has quotient $Q(x)$, then $Q(0) = f(0)$
(3) If $f(x)$ is divisible by $x + 1$, then $F(x) - F(0)$ is divisible by $(x+1)^{2}$
(4) If $F(x) \geq \frac{x^{2}}{2}$ holds for all real numbers $x$, then $f(x) \geq x$ also holds for all real numbers $x$
(5) If $f(x) \geq x$ holds for all $x > 0$, then $F(x) \geq \frac{x^{2}}{2}$ also holds for all $x > 0$