A digital black and white image is composed of small squares (pixels) whose colour ranges from white to black through all shades of grey. Each shade is coded by a real number $x$ as follows:

• $x = 0$ for white;
• $x = 1$ for black;
• $x = 0.01; x = 0.02$ and so on up to $x = 0.99$ in steps of 0.01 for all intermediate shades (from light to dark).

A function $f$ defined on the interval $[0; 1]$ is called a ``retouching function'' if it has the following four properties:

• $f(0) = 0$;
• $f(1) = 1$;
• $f$ is continuous on the interval $[0; 1]$;
• $f$ is increasing on the interval $[0; 1]$.

A shade coded $x$ is said to be darkened by the function $f$ if $f(x) > x$, and lightened if $f(x) < x$.

Part A

1. We consider the function $f_{1}$ defined on the interval $[0; 1]$ by: $$f_{1}(x) = 4x^{3} - 6x^{2} + 3x$$ a) Prove that the function $f_{1}$ is a retouching function. b) Solve graphically the inequality $f_{1}(x) \leq x$, using the graph given in the appendix, to be returned with your answer sheet, showing the necessary dotted lines. Interpret this result in terms of lightening or darkening.
2. We consider the function $f_{2}$ defined on the interval $[0; 1]$ by: $$f_{2}(x) = \ln[1 + (e - 1)x]$$ We admit that $f_{2}$ is a retouching function. We define on the interval $[0; 1]$ the function $g$ by: $g(x) = f_{2}(x) - x$. a) Establish that, for all $x$ in the interval $[0; 1]$: $g'(x) = \frac{(e - 2) - (e - 1)x}{1 + (e - 1)x}$; b) Determine the variations of the function $g$ on the interval $[0; 1]$. Prove that the function $g$ has a maximum at $\frac{e - 2}{e - 1}$, a maximum whose value rounded to the nearest hundredth is 0.12. c) Establish that the equation $g(x) = 0.05$ has two solutions $\alpha$ and $\beta$ on the interval $[0; 1]$, with $\alpha < \beta$. We will admit that: $0.08 < \alpha < 0.09$ and that: $0.85 < \beta < 0.86$.

Part B

We note that a modification of shade is visually perceptible only if the absolute value of the difference between the code of the initial shade and the code of the modified shade is greater than or equal to 0.05.

1. In the algorithm described below, $f$ denotes a retouching function. What is the role of this algorithm? \begin{verbatim} Variables : x (initial shade) y (retouched shade) E (difference) c (counter) k Initialization : c takes the value 0 Processing: For k ranging from 0 to 100, do x takes the value k/100 y takes the value f(x) E takes the value |y - x| If E >= 0.05, do c takes the value c + 1 End if End for Output: Display c \end{verbatim}
2. What value will this algorithm display if applied to the function $f_{2}$ defined in the second question of part $\mathbf{A}$?

Part C

In this part, we are interested in retouching functions $f$ whose effect is to lighten the image overall, that is, such that, for all real $x$ in the interval $[0; 1]$, $f(x) \leq x$. We decide to measure the overall lightening of the image by calculating the area $\mathscr{A}_{f}$ of the portion of the plane between the x-axis, the curve representing the function $f$, and the lines with equations $x = 0$ and $x = 1$ respectively. Between two functions, the one that has the effect of lightening the image the most is the one corresponding to the smallest area. We wish to compare the effect of the following two functions, which we admit are retouching functions:
$$f_{3}(x) = x\mathrm{e}^{(x^{2} - 1)} \quad f_{4}(x) = 4x - 15 + \frac{60}{x + 4}$$

1. a) Calculate $\mathscr{A}_{f_{3}}$. b) Calculate $\mathscr{A}_{f_{4}}$
2. Of these two functions, which one has the effect of lightening the image the most?

Let $f$ be the function defined on the interval $]0; +\infty[$ by: $$f(x) = x - 5\ln x - \frac{4}{x}$$

1. Determine the limit of $f(x)$ as $x$ tends to 0. You may use without proof the fact that $\lim_{x \rightarrow 0} x \ln x = 0$.
2. Determine the limit of $f(x)$ as $x$ tends to $+\infty$.
3. Prove that, for all strictly positive real $x$, $f'(x) = u(x)$, where $u(x) = \frac{x^2 - 5x + 4}{x^2}$.

Deduce the table of variations of the function $f$ by specifying the limits and particular values.

Let $f$ be the function defined on the interval $]0; +\infty[$ by $f(x) = \ln x$. For every strictly positive real number $a$, we define on $]0; +\infty[$ the function $g_a$ by $g_a(x) = ax^2$. We denote by $\mathscr{C}$ the curve representing the function $f$ and $\Gamma_a$ that of the function $g_a$ in a coordinate system of the plane. The purpose of the exercise is to study the intersection of the curves $\mathscr{C}$ and $\Gamma_a$ according to the values of the strictly positive real number $a$.
Part A
We have constructed in appendix 1 the curves $\mathscr{C}, \Gamma_{0,05}, \Gamma_{0,1}, \Gamma_{0,19}$ and $\Gamma_{0,4}$.

1. Name the different curves on the graph. No justification is required.
2. Use the graph to make a conjecture about the number of intersection points of $\mathscr{C}$ and $\Gamma_a$ according to the values (to be specified) of the real number $a$.

Part B
For a strictly positive real number $a$, we consider the function $h_a$ defined on the interval $]0; +\infty[$ by $$h_a(x) = \ln x - ax^2.$$

1. Justify that $x$ is the abscissa of a point $M$ belonging to the intersection of $\mathscr{C}$ and $\Gamma_a$ if and only if $h_a(x) = 0$.
2. a. We admit that the function $h_a$ is differentiable on $]0; +\infty[$ and we denote by $h_a'$ the derivative of the function $h_a$ on this interval. The variation table of the function $h_a$ is given below. Justify, by calculation, the sign of $h_a'(x)$ for $x$ belonging to $]0; +\infty[$.
   

   $x$	0		$\frac{1}{\sqrt{2a}}$
   $h_a'(x)$		+	0	-
   			$\frac{-1 - \ln(2a)}{2}$
   $h_a(x)$

   
   b. Recall the limit of $\frac{\ln x}{x}$ as $x \to +\infty$. Deduce the limit of the function $h_a$ as $x \to +\infty$. We do not ask you to justify the limit of $h_a$ at 0.
   
3. In this question and only in this question, we assume that $a = 0,1$. a. Justify that, in the interval $\left.]0; \frac{1}{\sqrt{0,2}}\right]$, the equation $h_{0,1}(x) = 0$ admits a unique solution. We admit that this equation also has only one solution in the interval $]\frac{1}{\sqrt{0,2}}; +\infty[$. b. What is the number of intersection points of $\mathscr{C}$ and $\Gamma_{0,1}$?
   
4. In this question and only in this question, we assume that $a = \frac{1}{2\mathrm{e}}$. a. Determine the value of the maximum of $h_{\frac{1}{2\mathrm{e}}}$. b. Deduce the number of intersection points of the curves $\mathscr{C}$ and $\Gamma_{\frac{1}{2\mathrm{e}}}$. Justify.
   
5. What are the values of $a$ for which $\mathscr{C}$ and $\Gamma_a$ have no intersection points? Justify.

For each real number $a$, we consider the function $f _ { a }$ defined on the set of real numbers $\mathbb { R }$ by
$$f _ { a } ( x ) = \mathrm { e } ^ { x - a } - 2 x + \mathrm { e } ^ { a } .$$

1. Show that for every real number $a$, the function $f _ { a }$ has a minimum.
2. Does there exist a value of $a$ for which this minimum is as small as possible?

Consider the function $f$ defined and differentiable on the interval $[0 ; +\infty[$ by
$$f ( x ) = x \mathrm { e } ^ { - x } - 0,1$$

1. Determine the limit of $f$ as $x \to +\infty$.
2. Study the variations of $f$ on $[0 ; +\infty[$ and draw the variation table.
3. Prove that the equation $f ( x ) = 0$ has a unique solution denoted $\alpha$ on the interval $[0 ; 1]$.

We admit the existence of a strictly positive real number $\beta$ such that $\alpha < \beta$ and $f ( \beta ) = 0$. We denote by $\mathscr { C }$ the representative curve of the function $f$ on the interval $[\alpha ; \beta]$ in an orthogonal coordinate system and $\mathscr { C } ^ { \prime }$ the curve symmetric to $\mathscr { C }$ with respect to the $x$-axis.
The unit on each axis represents 5 meters. These curves are used to delimit a floral bed in the shape of a candle flame on which tulips will be planted.

1. Prove that the function $F$, defined on the interval $[\alpha ; \beta]$ by $$F ( x ) = - ( x + 1 ) \mathrm { e } ^ { - x } - 0,1 x$$ is an antiderivative of the function $f$ on the interval $[\alpha ; \beta]$.
2. Calculate, in square units, a value rounded to 0.01 of the area of the region between the curves $\mathscr { C }$ and $\mathscr { C } ^ { \prime }$. Use the following values rounded to 0.001: $\alpha \approx 0.112$ and $\beta \approx 3.577$.
3. Knowing that 36 tulip plants can be placed per square meter, calculate the number of tulip plants needed for this floral bed.

Exercise 1 (5 points)
The Delmas chocolate factory decides to market new confectionery: chocolate drops in the shape of a water droplet. To do this, it must manufacture custom moulds that must meet the following constraint: for this range of sweets to be profitable, the chocolate factory must be able to produce at least 80 with 1 litre of liquid chocolate paste.

Part A: modelling by a function

The half-perimeter of the upper face of the drop will be modelled by a portion of the curve of the function $f$ defined on $]0;+\infty[$ by: $$f(x) = \frac{x^2 - 2x - 2 - 3\ln x}{x}.$$

1. Let $\varphi$ be the function defined on $]0;+\infty[$ by: $$\varphi(x) = x^2 - 1 + 3\ln x.$$ a. Calculate $\varphi(1)$ and the limit of $\varphi$ at 0. b. Study the variations of $\varphi$ on $]0;+\infty[$. Deduce the sign of $\varphi(x)$ according to the values of $x$.
   
2. a. Calculate the limits of $f$ at the boundaries of its domain of definition. b. Show that on $]0;+\infty[$: $f'(x) = \dfrac{\varphi(x)}{x^2}$. Deduce the variation table of $f$. c. Prove that the equation $f(x) = 0$ has a unique solution $\alpha$ on $]0;1]$. Determine using a calculator an approximate value of $\alpha$ to $10^{-2}$ near. It is admitted that the equation $f(x) = 0$ also has a unique solution $\beta$ on $[1;+\infty[$ with $\beta \approx 3.61$ to $10^{-2}$ near. d. Let $F$ be the function defined on $]0;+\infty[$ by: $$F(x) = \frac{1}{2}x^2 - 2x - 2\ln x - \frac{3}{2}(\ln x)^2.$$ Show that $F$ is an antiderivative of $f$ on $]0;+\infty[$.

Part B: solving the problem

In this part, calculations will be performed with the approximate values to $10^{-2}$ near of $\alpha$ and $\beta$ from Part A. To obtain the shape of the droplet, we consider the representative curve $C$ of the function $f$ restricted to the interval $[\alpha;\beta]$ as well as its reflection $C'$ with respect to the horizontal axis. The two curves $C$ and $C'$ delimit the upper face of the drop. For aesthetic reasons, the chocolatier would like his drops to have a thickness of $0.5$ cm. Under these conditions, would the profitability constraint be respected?

A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
Part B: study of functions

1. Let $f$ be the function defined on $]0; +\infty[$ by:
   $$f ( x ) = \frac { 105 } { x } \left( 1 - \mathrm { e } ^ { - \frac { 3 } { 40 } x } \right)$$
   Prove that, for every real $x$ in $]0; +\infty[$, $f ^ { \prime } ( x ) = \frac { 105 g ( x ) } { x ^ { 2 } }$, where $g$ is the function defined on $[0; +\infty[$ by:
   $$g ( x ) = \frac { 3 x } { 40 } \mathrm { e } ^ { - \frac { 3 } { 40 } x } + \mathrm { e } ^ { - \frac { 3 } { 40 } x } - 1$$
   
2. The variation table of the function $g$ is given:
   

   $x$	0	$+\infty$
   	0
   $g ( x )$		- 1

   
   Deduce the monotonicity of the function $f$. The limits of the function $f$ are not required.
   
3. Show that the equation $f ( x ) = 5.9$ has a unique solution on the interval $[1; 80]$. Deduce that this equation has a unique solution on the interval $]0; +\infty[$. Give an approximate value of this solution to the nearest tenth.

In a cardboard disk of radius $R$, we cut out an angular sector corresponding to an angle of measure $\alpha$ radians. We overlap the edges to create a cone of revolution. We wish to choose the angle $\alpha$ to obtain a cone of maximum volume.
We call $\ell$ the radius of the circular base of this cone and $h$ its height. We recall that:

• the volume of a cone of revolution with base a disk of area $\mathscr{A}$ and height $h$ is $\frac{1}{3}\mathscr{A}h$.
• the length of an arc of a circle of radius $r$ and angle $\theta$, expressed in radians, is $r\theta$.

1. We choose $R = 20\mathrm{~cm}$. a. Show that the volume of the cone, as a function of its height $h$, is $$V(h) = \frac{1}{3}\pi\left(400 - h^2\right)h.$$ b. Justify that there exists a value of $h$ that makes the volume of the cone maximum. Give this value. c. How should we cut the cardboard disk to have maximum volume? Give an approximation of $\alpha$ to the nearest degree.
2. Does the angle $\alpha$ depend on the radius $R$ of the cardboard disk?

Let $k$ be a strictly positive real number. Consider the functions $f _ { k }$ defined on $\mathbb { R }$ by: $$f _ { k } ( x ) = x + k \mathrm { e } ^ { - x } .$$ We denote by $\mathscr { C } _ { k }$ the representative curve of function $f _ { k }$ in a plane with an orthonormal coordinate system.
For every strictly positive real number $k$, the function $f _ { k }$ admits a minimum on $\mathbb { R }$. The value at which this minimum is attained is the abscissa of the point denoted $A _ { k }$ on the curve $\mathscr { C } _ { k }$. It would seem that, for every strictly positive real number $k$, the points $A _ { k }$ are collinear. Is this the case?

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: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ In this question, round both results to the nearest thousandth. a. Calculate $f(20)$. b. Determine the maximum rate of $\mathrm{CO}_2$ present in the room during the experiment.

Exercise 2

During a laboratory experiment, a projectile is launched into a fluid medium. The objective is to determine for which firing angle $\theta$ with respect to the horizontal the height of the projectile does not exceed 1.6 meters. Since the projectile does not move through air but through a fluid, the usual parabolic model is not adopted. Here we model the projectile as a point that moves, in a vertical plane, on the curve representing the function $f$ defined on the interval $[0; 1[$ by: $$f(x) = bx + 2\ln(1-x)$$ where $b$ is a real parameter greater than or equal to 2, $x$ is the abscissa of the projectile, $f(x)$ its ordinate, both expressed in meters.

1. The function $f$ is differentiable on the interval $[0; 1[$. We denote $f'$ its derivative function.
   We admit that the function $f$ has a maximum on the interval $[0; 1[$ and that, for every real $x$ in the interval $[0; 1[$: $$f'(x) = \frac{-bx + b - 2}{1 - x}$$ Show that the maximum of the function $f$ is equal to $b - 2 + 2\ln\left(\frac{2}{b}\right)$.
2. Determine for which values of the parameter $b$ the maximum height of the projectile does not exceed 1.6 meters.
3. In this question, we choose $b = 5.69$.
   The firing angle $\theta$ corresponds to the angle between the abscissa axis and the tangent to the curve of the function $f$ at the point with abscissa 0. Determine an approximate value of the angle $\theta$ to the nearest tenth of a degree.

Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
It is admitted that, for all real $a$ in the interval $]0; 1]$, the area of triangle $\mathrm{O}N_aP_a$ in square units is given by $\mathscr{A}(a) = \frac{1}{2}a(1 - \ln a)^2$.
Using the previous questions, determine for which value of $a$ the area $\mathscr{A}(a)$ is maximum. Determine this maximum area.

Part A
Let $g$ be the function defined on the set of real numbers $\mathbf { R }$, by $$g ( x ) = x ^ { 2 } + x + \frac { 1 } { 4 } + \frac { 4 } { \left( 1 + \mathrm { e } ^ { x } \right) ^ { 2 } }$$
It is admitted that the function $g$ is differentiable on $\mathbf { R }$ and we denote by $g ^ { \prime }$ its derivative function.
1. Determine the limits of $g$ at $+ \infty$ and at $- \infty$.
2. It is admitted that the function $g ^ { \prime }$ is strictly increasing on $\mathbf { R }$ and that $g ^ { \prime } ( 0 ) = 0$.
Determine the sign of the function $g ^ { \prime }$ on $\mathbf { R }$.
3. Draw up the table of variations of the function $g$ and calculate the minimum of the function $g$ on $\mathbf { R }$.
Part B
Let $f$ be the function defined on $\mathbf { R }$ by: $$f ( x ) = 3 - \frac { 2 } { 1 + \mathrm { e } ^ { x } }$$
We denote by $\mathscr { C } _ { f }$ the representative curve of $f$ in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).
Let A be the point with coordinates $\left( - \frac { 1 } { 2 } ; 3 \right)$.
1. Prove that point $\mathrm { B } ( 0 ; 2 )$ belongs to $\mathscr { C } _ { f }$.
2. Let $x$ be any real number. We denote by $M$ the point on the curve $\mathscr { C } _ { f }$ with coordinates $( x ; f ( x ) )$.
Prove that $\mathrm { A } M ^ { 2 } = g ( x )$.
3. It is admitted that the distance $\mathrm { A } M$ is minimal if and only if $\mathrm { A } M ^ { 2 }$ is minimal.
Determine the coordinates of the point on the curve $\mathscr { C } _ { f }$ such that the distance AM is minimal.
4. It is admitted that the function $f$ is differentiable on $\mathbf { R }$ and we denote by $f ^ { \prime }$ its derivative function.
a. Calculate $f ^ { \prime } ( x )$ for all real $x$.
b. Let $T$ be the tangent to the curve $\mathscr { C } _ { f }$ at point B.
Prove that the reduced equation of $T$ is $y = \frac { x } { 2 } + 2$.
5. Prove that the line $T$ is perpendicular to the line (AB).

Exercise B (Main topics: Sequences, function study, Logarithm function)
Let the function $f$ defined on the interval $]1; +\infty[$ by $$f(x) = x - \ln(x-1).$$ Consider the sequence $(u_n)$ with initial term $u_0 = 10$ and such that $u_{n+1} = f(u_n)$ for every natural integer $n$.

Part I:

The spreadsheet below was used to obtain approximate values of the first terms of the sequence $(u_n)$.

	A	B
1	$n$	$u_n$
2	0	10
3	1	7.80277542
4	2	5.88544474
5	3	4.29918442
6	4	3.10550913
7	5	2.36095182
8	6	2.0527675
9	7	2.00134509
10	8	2.0000009

1. What formula was entered in cell B3 to allow the calculation of approximate values of $(u_n)$ by copying downward?
2. Using these values, conjecture the direction of variation and the limit of the sequence $(u_n)$.

Part II:

We recall that the function $f$ is defined on the interval $]1; +\infty[$ by $$f(x) = x - \ln(x-1).$$

1. Calculate $\lim_{x \rightarrow 1} f(x)$. We will admit that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.
2. a. Let $f^{\prime}$ be the derivative function of $f$. Show that for all $x \in ]1; +\infty[$, $f^{\prime}(x) = \frac{x-2}{x-1}$. b. Deduce the table of variations of $f$ on the interval $]1; +\infty[$, completed by the limits. c. Justify that for all $x \geqslant 2$, $f(x) \geqslant 2$.

Part III:

1. Using the results of Part II, prove by induction that $u_n \geqslant 2$ for every natural integer $n$.
2. Show that the sequence $(u_n)$ is decreasing.
3. Deduce that the sequence $(u_n)$ is convergent. We denote its limit by $\ell$.
4. We admit that $\ell$ satisfies $f(\ell) = \ell$. Give the value of $\ell$.

Let $f$ be the function defined on the interval $]0;+\infty[$ by: $$f(x) = \frac{\mathrm{e}^{x}}{x}.$$ We denote $\mathscr{C}_{f}$ the representative curve of the function $f$ in an orthonormal coordinate system.

1. a. Specify the limit of the function $f$ at $+\infty$. b. Justify that the $y$-axis is an asymptote to the curve $\mathscr{C}_{f}$.
2. Show that, for every real number $x$ in the interval $]0;+\infty[$, we have: $$f^{\prime}(x) = \frac{\mathrm{e}^{x}(x-1)}{x^{2}}$$ where $f^{\prime}$ denotes the derivative function of the function $f$.
3. Determine the variations of the function $f$ on the interval $]0;+\infty[$. A variation table of the function $f$ will be established in which the limits appear.
4. Let $m$ be a real number. Specify, depending on the values of the real number $m$, the number of solutions of the equation $f(x) = m$.
5. We denote $\Delta$ the line with equation $y = -x$.
   We denote A a possible point of $\mathscr{C}_{f}$ with abscissa $a$ at which the tangent to the curve $\mathscr{C}_{f}$ is parallel to the line $\Delta$. a. Show that $a$ is a solution of the equation $\mathrm{e}^{x}(x-1) + x^{2} = 0$.
   We denote $g$ the function defined on $[0;+\infty[$ by $g(x) = \mathrm{e}^{x}(x-1) + x^{2}$. We assume that the function $g$ is differentiable and we denote $g^{\prime}$ its derivative function. b. Calculate $g^{\prime}(x)$ for every real number $x$ in the interval $[0;+\infty[$, then establish the variation table of $g$ on $[0;+\infty[$. c. Show that there exists a unique point $A$ at which the tangent to $\mathscr{C}_{f}$ is parallel to the line $\Delta$.

Below is the graphical representation of $f^{\prime}$, the derivative function of a function $f$ defined on [0;7].
The variation table of $f$ on the interval [0; 7] is:
a.

$x$	0	3,25	7
$f(x)$

b.

$x$	0	2	5	7
$f(x)$

c.

$x$	0	2	5	7
$f(x)$	$\nearrow$

d.

$x$	0	2	7
$f(x)$

Part I

We consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = x - \mathrm{e}^{-2x}$$
We call $\Gamma$ the representative curve of the function $f$ in an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath})$.

1. Determine the limits of the function $f$ at $-\infty$ and at $+\infty$.
2. Study the monotonicity of the function $f$ on $\mathbb{R}$ and draw up its variation table.
3. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on $\mathbb{R}$, and give an approximate value to $10^{-2}$ precision.
4. Deduce from the previous questions the sign of $f(x)$ according to the values of $x$.

Part II

In the orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath})$, we call $\mathscr{C}$ the representative curve of the function $g$ defined on $\mathbb{R}$ by:
$$g(x) = \mathrm{e}^{-x}$$
The curves $\mathscr{C}$ and the curve $\Gamma$ (which represents the function $f$ from Part I) are drawn on the graph provided in the appendix which is to be completed and returned with your paper. The purpose of this part is to determine the point on the curve $\mathscr{C}$ closest to the origin O of the coordinate system and to study the tangent to $\mathscr{C}$ at this point.

1. For any real number $t$, we denote by $M$ the point with coordinates $(t; \mathrm{e}^{-t})$ on the curve $\mathscr{C}$.
   We consider the function $h$ which, to the real number $t$, associates the distance $OM$. We therefore have: $h(t) = OM$, that is:
   $$h(t) = \sqrt{t^2 + \mathrm{e}^{-2t}}$$
   a. Show that, for any real number $t$,
   $$h'(t) = \frac{f(t)}{\sqrt{t^2 + \mathrm{e}^{-2t}}}$$
   where $f$ denotes the function studied in Part I. b. Prove that the point A with coordinates $(\alpha; \mathrm{e}^{-\alpha})$ is the point on the curve $\mathscr{C}$ for which the length $OM$ is minimal. Place this point on the graph provided in the appendix, to be returned with your paper.
   
2. We call $T$ the tangent to the curve $\mathscr{C}$ at A. a. Express in terms of $\alpha$ the slope of the tangent $T$.
   We recall that the slope of the line (OA) is equal to $\frac{\mathrm{e}^{-\alpha}}{\alpha}$. We also recall the following result which may be used without proof: In an orthonormal coordinate system of the plane, two lines $D$ and $D'$ with slopes $m$ and $m'$ respectively are perpendicular if and only if the product $mm'$ is equal to $-1$. b. Prove that the line (OA) and the tangent $T$ are perpendicular.
   Draw these lines on the graph provided in the appendix, to be returned with your paper.

We are given a function $f$, assumed to be differentiable on $\mathbb{R}$, and we denote $f^{\prime}$ its derivative function.
Below is the variation table of $f$:

$x$	$-\infty$	$-1$	$+\infty$
$f(x)$
	$-\infty$	0

According to this variation table: a. $f^{\prime}$ is positive on $\mathbb{R}$. b. $f^{\prime}$ is positive on $\left.]-\infty;-1\right]$ c. $f^{\prime}$ is negative on $\mathbb{R}$ d. $f^{\prime}$ is positive on $[-1;+\infty[$

We consider the function $f$ defined on $]0; 8]$ by $$f ( x ) = \frac { 10 \ln \left( - x ^ { 2 } + 7 x + 9 \right) } { x }$$ Let $C _ { f }$ be the graphical representation of the function $f$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

Part A

1. Solve in $\mathbb { R }$ the inequality $- x ^ { 2 } + 7 x + 8 \geqslant 0$.
2. Deduce that for all $x \in ] 0 ; 8 ]$, we have $f ( x ) \geqslant 0$.
3. Interpret this result graphically.

Part B

The curve $C _ { f }$ is represented below. Let $M$ be the point of $C _ { f }$ with abscissa $x$ where $x \in ] 0; 8]$. We call $N$ and $P$ the orthogonal projections of the point $M$ respectively on the abscissa axis and on the ordinate axis. In this part, we are interested in the area $\mathscr { A } ( x )$ of the rectangle $\mathrm{O}NMP$.

1. Give the coordinates of points $N$ and $P$ as a function of $x$.
2. Show that for all $x$ belonging to the interval $] 0 ; 8 ]$, $$\mathscr { A } ( x ) = 10 \ln \left( - x ^ { 2 } + 7 x + 9 \right)$$
3. Does there exist a position of the point $M$ for which the area of the rectangle $\mathrm{O}NMP$ is maximum? If it exists, determine this position.

Part C

We consider a strictly positive real number $k$. We wish to determine the smallest value of $x$, approximated to the nearest tenth, belonging to $[ 3.5; 8 ]$ for which the area $\mathscr { A } ( x )$ becomes less than or equal to $k$. To do this, we consider the algorithm below. As a reminder, in Python language, $\ln ( x )$ is written log$(x)$.
\begin{verbatim} from math import * def A(x) : return 10*log (- 1* x**2 + 7*x + 9) def pluspetitevaleur(k) : x = 3.5 while A(x).........: x = x + 0.1 return ........... \end{verbatim}

1. Copy and complete lines 8 and 10 of the algorithm.
2. What number does the instruction \texttt{pluspetitevaleur(30)} then return?
3. What happens when $k = 35$? Justify.

Having a large piece of land, an entertainment company intends to build a rectangular space for shows and events, as shown in the figure.
The area for the public will be fenced with two types of materials:

• on the sides parallel to the stage, a type A screen will be used, more resistant, whose value per linear meter is $\mathrm{R}\$ 20.00$;
• on the other two sides, a type B screen will be used, common, whose linear meter costs $\mathrm{R}\$ 5.00$.

The company has $\mathrm{R}\$ 5000.00$ to buy all the screens, but wants to do it in such a way that it obtains the largest possible area for the public.
The quantity of each type of screen that the company should buy is
(A) $50.0 \mathrm{~m}$ of type A screen and $800.0 \mathrm{~m}$ of type B screen.
(B) $62.5 \mathrm{~m}$ of type A screen and $250.0 \mathrm{~m}$ of type B screen.
(C) $100.0 \mathrm{~m}$ of type A screen and $600.0 \mathrm{~m}$ of type B screen.
(D) $125.0 \mathrm{~m}$ of type A screen and $500.0 \mathrm{~m}$ of type B screen.
(E) $200.0 \mathrm{~m}$ of type A screen and $200.0 \mathrm{~m}$ of type B screen.

Lobster hatcheries are built, by local fishing cooperatives, in the shape of right-rectangular prisms, fixed to the ground and with flexible nets of the same height, capable of withstanding marine corrosion. For each hatchery to be built, the cooperative uses entirely 100 linear meters of this net, which is used only on the sides.
What should be the values of $X$ and $Y$, in meters, so that the area of the base of the hatchery is maximum?
(A) 1 and 49
(B) 1 and 99
(C) 10 and 10
(D) 25 and 25
(E) 50 and 50

$f ( x ) = x ^ { 3 } + x ^ { 2 } + c x + d$, where $c$ and $d$ are real numbers. Prove that if $c > \frac { 1 } { 3 }$, then $f$ has exactly one real root.

Consider the function $f ( x ) = a x + \frac { 1 } { x + 1 }$, where $a$ is a positive constant. Let $L =$ the largest value of $f ( x )$ and $S =$ the smallest value of $f ( x )$ for $x \in [ 0,1 ]$. Show that $L - S > \frac { 1 } { 12 }$ for any $a > 0$.

Let $f ( x ) = ( x - a ) ( x - b ) ^ { 3 } ( x - c ) ^ { 5 } ( x - d ) ^ { 7 }$, where $a , b , c , d$ are real numbers with $a < b < c < d$. Thus $f ( x )$ has 16 real roots counting multiplicities and among them 4 are distinct from each other. Consider $f ^ { \prime } ( x )$, i.e. the derivative of $f ( x )$. Find the following, if you can: (i) the number of real roots of $f ^ { \prime } ( x )$, counting multiplicities, (ii) the number of distinct real roots of $f ^ { \prime } ( x )$.

By definition the region inside the parabola $y = x^{2}$ is the set of points $(a,b)$ such that $b \geq a^{2}$. We are interested in those circles all of whose points are in this region. A bubble at a point $P$ on the graph of $y = x^{2}$ is the largest such circle that contains $P$. (You may assume the fact that there is a unique such circle at any given point on the parabola.)
(a) A bubble at some point on the parabola has radius 1. Find the center of this bubble.
(b) Find the radius of the smallest possible bubble at some point on the parabola. Justify.