9. Which of the following pairs of graphs could represent the graph of a function and the graph of its derivative? I. [Figure] [Figure] II. [Figure] [Figure] III. [Figure] [Figure]
(A) I only
(B) II only
(C) III only
(D) I and III
(E) II and III

The graph of $f ^ { \prime }$, the derivative of the function $f$, is shown above. Which of the following could be the graph of $f$?
(A), (B), (C), (D) [graphs as shown in the exam]

The graph of the function $f$ is shown in the figure above. The value of $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ is
(A) - 2
(B) - 1
(C) 2
(D) nonexistent

The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

The graph of the function $f$ is shown in the figure above. For which of the following values of $x$ is $f ^ { \prime } ( x )$ positive and increasing?
(A) $a$
(B) $b$
(C) $c$
(D) $d$
(E) $e$

The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

The derivative of a function $f$ is increasing for $x < 0$ and decreasing for $x > 0$. Which of the following could be the graph of $f$ ?
(A), (B), (C), (D), (E) [graphs shown in figures above]

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?

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?

The curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ given in appendix 1 are the graphical representations, in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ), of two functions $f$ and $g$ defined on $[ 0 ; + \infty [$. We consider the points $\mathrm { A } ( 0,5 ; 1 )$ and $\mathrm { B } ( 0 ; - 1 )$ in the coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ). We know that O belongs to $\mathscr { C } _ { f }$ and that the line (OA) is tangent to $\mathscr { C } _ { f }$ at point O.

1. We assume that the function $f$ is written in the form $f ( x ) = ( a x + b ) \mathrm { e } ^ { - x ^ { 2 } }$ where $a$ and $b$ are real numbers. Determine the exact values of the real numbers $a$ and $b$, detailing the approach. From now on, we consider that $\boldsymbol { f } ( \boldsymbol { x } ) = \mathbf { 2 } \boldsymbol { x } \mathrm { e } ^ { - \boldsymbol { x } ^ { \mathbf { 2 } } }$ for all $\boldsymbol { x }$ belonging to $[ \mathbf { 0 } ; + \infty [$
2. a. We will admit that, for all real $x$ strictly positive, $f ( x ) = \frac { 2 } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$.
   Calculate $\lim _ { x \rightarrow + \infty } f ( x )$. b. Draw up, justifying it, the table of variations of the function $f$ on $[ 0 ; + \infty [$.
3. The function $g$ whose representative curve $\mathscr { C } _ { g }$ passes through the point $\mathrm { B } ( 0 ; - 1 )$ is a primitive of the function $f$ on $[ 0 ; + \infty [$. a. Determine the expression of $g ( x )$. b. Let $m$ be a strictly positive real number.
   Calculate $I _ { m } = \int _ { 0 } ^ { m } f ( t ) \mathrm { d } t$ as a function of $m$. c. Determine $\lim _ { m \rightarrow + \infty } I _ { m }$.
4. a. Justify that $f$ is a probability density function on $[ 0 ; + \infty [$. b. Let $X$ be a continuous random variable that admits the function $f$ as its probability density function. Justify that, for all real $x$ in $[ 0 ; + \infty [$, $P ( X \leqslant x ) = g ( x ) + 1$. c. Deduce the exact value of the real number $\alpha$ such that $P ( X \leqslant \alpha ) = 0,5$. d. Without using an approximate value of $\alpha$, construct in the coordinate system of appendix 1 the point with coordinates ( $\alpha ; 0$ ) leaving the construction lines visible. Then shade the region of the plane corresponding to $P ( X \leqslant \alpha )$.

Exercise 1 - Part A

Here are two curves $\mathcal { C } _ { 1 }$ and $\mathcal { C } _ { 2 }$ which give for two people $P _ { 1 }$ and $P _ { 2 }$ of different body compositions the concentration $C$ of alcohol in the blood (blood alcohol level) as a function of time $t$ after ingestion of the same quantity of alcohol. The instant $t = 0$ corresponds to the moment when the two individuals ingest the alcohol. $C$ is expressed in grams per litre and $t$ in hours.

1. The function $C$ is defined on the interval $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $C ^ { \prime }$ its derivative function. At an instant $t$ positive or zero, the rate of appearance of alcohol in the blood is given by $C ^ { \prime } ( t )$. At what instant is this rate maximal? It is often said that a person of weak body composition experiences the effects of alcohol more quickly.
2. On the previous graph, identify the curve corresponding to the person with the largest body composition. Justify the choice made.
3. A person on an empty stomach ingests alcohol. It is admitted that the concentration $C$ of alcohol in their blood can be modelled by the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( t ) = A t \mathrm { e } ^ { - t }$$ where $A$ is a positive constant that depends on the body composition and the quantity of alcohol ingested. a. We denote $f ^ { \prime }$ the derivative function of the function $f$. Determine $f ^ { \prime } ( 0 )$. b. Is the following statement true? ``For equal quantities of alcohol ingested, the larger $A$ is, the more corpulent the person is.''

Part B - A particular case

Paul, a 19-year-old student of average body composition and a young driver, drinks two glasses of rum. The concentration $C$ of alcohol in his blood is modelled as a function of time $t$, expressed in hours, by the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( t ) = 2 t \mathrm { e } ^ { - t } .$$

1. Study the variations of the function $f$ on the interval $[ 0 ; + \infty [$.
2. At what instant is the concentration of alcohol in Paul's blood maximal? What is its value then? Round to $10 ^ { - 2 }$ near.
3. Recall the limit of $\frac { \mathrm { e } ^ { t } } { t }$ as $t$ tends to $+ \infty$ and deduce from it that of $f ( t )$ at $+ \infty$. Interpret the result in the context of the exercise.
4. Paul wants to know after how much time he can take his car. We recall that the legislation allows a maximum concentration of alcohol in the blood of $0,2 \mathrm {~g} . \mathrm { L } ^ { - 1 }$ for a young driver. a. Prove that there exist two real numbers $t _ { 1 }$ and $t _ { 2 }$ such that $f \left( t _ { 1 } \right) = f \left( t _ { 2 } \right) = 0,2$. b. What minimum duration must Paul wait before he can take the wheel in full compliance with the law? Give the result rounded to the nearest minute.
5. The minimum concentration of alcohol detectable in the blood is estimated at $5 \times 10 ^ { - 3 }$ g.L${}^{ - 1 }$. a. Justify that there exists an instant $T$ from which the concentration of alcohol in the blood is no longer detectable. b. The following algorithm is given where $f$ is the function defined by $f ( t ) = 2 t \mathrm { e } ^ { - t }$.
   

   Initialization:	$t$ takes the value 3,5
   	$p$ takes the value 0,25
   	$C$ takes the value 0,21
   Processing:	While $C > 5 \times 10 ^ { - 3 }$ do:
   	$\quad \mid \quad t$ takes the value $t + p$
   	$\quad C$ takes the value $f ( t )$
   Output:	End While
   Display $t$

   
   Copy and complete the following table of values by executing this algorithm. Round the values to $10 ^ { - 2 }$ near.
   

   	Initialization	Step 1	Step 2
   $p$	0,25
   $t$	3,5
   $C$	0,21

   
   What does the value displayed by this algorithm represent?

We consider the function $g$ defined for all real $x$ by $g(x) = \mathrm{e}^{1-x}$. The representative curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ respectively are drawn in a coordinate system. The purpose of this part is to study the relative position of these two curves.

1. After observing the graph, what conjecture can be made?
2. Justify that, for all real $x$ belonging to $]-\infty; 0]$, $f(x) < g(x)$.
3. In this question, we consider the interval $]0; +\infty[$. We set, for all strictly positive real $x$, $\Phi(x) = \ln x - x^{2} + x$. a. Show that, for all strictly positive real $x$, $$f(x) \leqslant g(x) \text{ is equivalent to } \Phi(x) \leqslant 0$$ It is admitted for the rest that $f(x) = g(x)$ is equivalent to $\Phi(x) = 0$. b. It is admitted that the function $\Phi$ is differentiable on $]0; +\infty[$. Draw the table of variations of the function $\Phi$. (The limits at 0 and $+\infty$ are not required.) c. Deduce that, for all strictly positive real $x$, $\Phi(x) \leqslant 0$.
4. a. Is the conjecture made in question 1 of Part B valid? b. Show that $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ have a unique common point, denoted $A$. c. Show that at this point $A$, these two curves have the same tangent line.

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
$f(x)$

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 $[-1;+\infty[$

Part A

In the orthonormal coordinate system above, the representative curves of a function $f$ and its derivative function, denoted $f ^ { \prime }$, are drawn, both defined on $] 3 ; + \infty [$.

1. Associate each curve with the function it represents. Justify.
2. Determine graphically the possible solution(s) of the equation $f ( x ) = 3$.
3. Indicate, by graphical reading, the convexity of the function $f$.

Part B

1. Justify that the quantity $\ln \left( x ^ { 2 } - x - 6 \right)$ is well defined for values $x$ in the interval ]3; $+ \infty$ [, which we will call $I$ in the following.
2. We admit that the function $f$ from Part A is defined by $f ( x ) = \ln \left( x ^ { 2 } - x - 6 \right)$ on $I$. Calculate the limits of the function $f$ at the two endpoints of the interval $I$. Deduce an equation of an asymptote to the representative curve of the function $f$ on $I$.
3. a. Calculate $f ^ { \prime } ( x )$ for all $x$ belonging to $I$. b. Study the direction of variation of the function $f$ on $I$.

Draw the variation table of the function $f$ showing the limits at the endpoints of $I$.
4. a. Justify that the equation $f ( x ) = 3$ admits a unique solution $\alpha$ on the interval ]5; 6[. b. Determine, using a calculator, an approximation of $\alpha$ to within $10 ^ { - 2 }$.
5. a. Justify that $f ^ { \prime \prime } ( x ) = \frac { - 2 x ^ { 2 } + 2 x - 13 } { \left( x ^ { 2 } - x - 6 \right) ^ { 2 } }$. b. Study the convexity of the function $f$ on $I$.

Consider a function $f$ defined and twice differentiable on $\mathbb { R }$. We call $\mathscr { C }$ its graphical representation. We denote by $f ^ { \prime \prime }$ the second derivative of $f$. The curve of $f ^ { \prime \prime }$, denoted $\mathscr { C } ^ { \prime \prime }$, is represented in the graph opposite. a. $\mathscr { C }$ admits a unique inflection point; b. $f$ is convex on the interval $[ - 1 ; 2 ]$; c. $f$ is convex on $] - \infty ; - 1 ]$ and on $[2; + \infty [$; d. $f$ is convex on $\mathbb { R }$.

Exercise 4 (7 points) Theme: numerical functions This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The six questions are independent.

1. The representative curve of the function $f$ defined on $\mathbb{R}$ by $f(x) = \frac{-2x^2 + 3x - 1}{x^2 + 1}$ admits as an asymptote the line with equation: a. $x = -2$; b. $y = -1$; c. $y = -2$; d. $y = 0$
2. Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = x\mathrm{e}^{x^2}$. The antiderivative $F$ of $f$ on $\mathbb{R}$ which satisfies $F(0) = 1$ is defined by: a. $F(x) = \frac{x^2}{2}\mathrm{e}^{x^2}$; b. $F(x) = \frac{1}{2}\mathrm{e}^{x^2}$ c. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2}$; d. $F(x) = \frac{1}{2}\mathrm{e}^{x^2} + \frac{1}{2}$
3. The representative graph $\mathscr{C}_{f'}$ of the derivative function $f'$ of a function $f$ defined on $\mathbb{R}$ is given below. We can affirm that the function $f$ is: a. concave on $]0; +\infty[$; b. convex on $]0; +\infty[$; c. convex on $[0; 2]$; d. convex on $[2; +\infty[$.
4. Among the antiderivatives of the function $f$ defined on $\mathbb{R}$ by $f(x) = 3\mathrm{e}^{-x^2} + 2$: a. all are increasing on $\mathbb{R}$; b. all are decreasing on $\mathbb{R}$; c. some are increasing on $\mathbb{R}$ and others decreasing on $\mathbb{R}$; d. all are increasing on $]-\infty; 0]$ and decreasing on $[0; +\infty[$.
5. The limit at $+\infty$ of the function $f$ defined on the interval $]0; +\infty[$ by $f(x) = \frac{2\ln x}{3x^2 + 1}$ is equal to: a. $\frac{2}{3}$; b. $+\infty$; c. $-\infty$; d. 0.
6. The equation $\mathrm{e}^{2x} + \mathrm{e}^x - 12 = 0$ admits in $\mathbb{R}$: a. three solutions; b. two solutions; c. only one solution; d. no solution.

The plane is equipped with an orthogonal coordinate system. We consider a function $f$ defined and differentiable on $\mathbb{R}$. We denote $f^{\prime}$ its derivative function. The representative curve of the derivative function $f^{\prime}$ is given.
In this part, results will be obtained by graphical reading of the representative curve of the derivative function $f^{\prime}$. No justification is required.

1. Give the direction of variation of the function $f$ on $\mathbb{R}$. Use approximate values if necessary.
2. Give the intervals on which the function $f$ appears to be convex.

Exercise 3 — 5 points Theme: function study Parts A and B can be treated independently
Part A
The plane is equipped with an orthogonal coordinate system. Below is represented the curve of a function $f$ defined and twice differentiable on $\mathbb{R}$, as well as that of its derivative $f'$ and its second derivative $f''$.

1. Determine, by justifying your choice, which curve corresponds to which function.
2. Determine, with the precision allowed by the graph, the slope of the tangent line to curve $\mathscr{C}_{2}$ at the point with abscissa 4.
3. Give, with the precision allowed by the graph, the abscissa of each inflection point of curve $\mathscr{C}_{1}$.

Part B
Let $k$ be a strictly positive real number. We consider the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \frac{4}{1 + \mathrm{e}^{-kx}}$$

1. Determine the limits of $g$ at $+\infty$ and at $-\infty$.
2. Prove that $g'(0) = k$.
3. By admitting the result below obtained with computer algebra software, prove that the curve of $g$ has an inflection point at the point with abscissa 0.

$\triangleright$	Computer algebra
	$g(x) = 4 / (1 + \mathrm{e}^{\wedge}(-kx))$
1
	$\rightarrow g(x) = \frac{4}{\mathrm{e}^{-kx} + 1}$
	Simplify $(g''(x))$
2
	$\rightarrow g''(x) = -4\mathrm{e}^{kx}(\mathrm{e}^{kx} - 1)\frac{k^{2}}{(\mathrm{e}^{kx} + 1)^{3}}$

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. No justification is required. A wrong answer, no answer, or multiple answers, neither gives nor takes away points.

1. We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = 2x\mathrm{e}^x$.
   The number of solutions on $\mathbb{R}$ of the equation $f(x) = -\dfrac{73}{100}$ is equal to:
   a. 0
   b. 1
   c. 2
   d. infinitely many.
   
2. We consider the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \frac{x+1}{\mathrm{e}^x}.$$ The limit of the function $g$ at $-\infty$ is equal to:
   a. $-\infty$
   b. $+\infty$
   c. $0$
   d. it does not exist.
   
3. We consider the function $h$ defined on $\mathbb{R}$ by: $$h(x) = (4x - 16)\mathrm{e}^{2x}.$$ We denote $\mathscr{C}_h$ the representative curve of $h$ in an orthogonal coordinate system. We can affirm that:
   a. $h$ is convex on $\mathbb{R}$.
   b. $\mathscr{C}_h$ has an inflection point at $x = 3$.
   c. $h$ is concave on $\mathbb{R}$.
   d. $\mathscr{C}_h$ has an inflection point at $x = 3.5$.
   
4. We consider the function $k$ defined on the interval $]0; +\infty[$ by: $$k(x) = 3\ln(x) - x.$$ We denote $\mathscr{C}$ the representative curve of the function $k$ in an orthonormal coordinate system. We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point with abscissa $x = \mathrm{e}$. An equation of $T$ is:
   a. $y = (3 - \mathrm{e})x$
   b. $y = \left(\dfrac{3 - \mathrm{e}}{\mathrm{e}}\right)x$
   c. $y = \left(\dfrac{3}{\mathrm{e}} - 1\right)x + 1$
   d. $y = (\mathrm{e} - 1)x + 1$
   
5. We consider the equation $[\ln(x)]^2 + 10\ln(x) + 21 = 0$, with $x \in ]0; +\infty[$.
   The number of solutions of this equation is equal to:
   a. 0
   b. 1
   c. 2
   d. infinitely many.

We consider a function $f$ defined on $[0; +\infty[$, represented by the curve $\mathscr{C}$ below. The line $T$ is tangent to the curve $\mathscr{C}$ at point A with abscissa $\frac{5}{2}$.

1. Draw up, by graphical reading, the table of variations of the function $f$ on the interval $[0;5]$.
2. What does the curve $\mathscr{C}$ appear to present at point A?
3. The derivative $f'$ and the second derivative $f''$ of the function $f$ are represented by the curves $\mathscr{C}_1$ and $\mathscr{C}_2$. Associate with each of these two functions the curve that represents it. This choice will be justified.
4. Can the curve $\mathscr{C}_3$ be the graphical representation on $[0; +\infty[$ of a primitive of the function $f$? Justify.

Part 1
We consider the function $f$ defined on the set of real numbers $\mathbb{R}$ by: $$f(x) = \left(x^2 - 4\right)\mathrm{e}^{-x}$$ We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function.

1. Determine the limits of the function $f$ at $-\infty$ and at $+\infty$.
2. Justify that for all real $x$, $f'(x) = \left(-x^2 + 2x + 4\right)\mathrm{e}^{-x}$.
3. Deduce the variations of the function $f$ on $\mathbb{R}$.

Part 2
We consider the sequence $(I_n)$ defined for all natural integer $n$ by $I_n = \int_{-2}^{0} x^n \mathrm{e}^{-x}\,\mathrm{d}x$.

1. Justify that $I_0 = \mathrm{e}^2 - 1$.
2. Using integration by parts, demonstrate the equality: $$I_{n+1} = (-2)^{n+1}\mathrm{e}^2 + (n+1)I_n$$
3. Deduce the exact values of $I_1$ and $I_2$.

Part 3

1. Determine the sign on $\mathbb{R}$ of the function $f$ defined in Part 1.
2. The curve $\mathscr{C}_f$ of the function $f$ is represented in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. The domain $D$ of the shaded region is bounded by the curve $\mathscr{C}_f$, the $x$-axis and the $y$-axis. Calculate the exact value, in square units, of the area $S$ of the domain $D$.

3. We admit that for all $x$ belonging to $] 0$; $+ \infty \left[ , f ^ { \prime } ( x ) = 2 ( \ln x ) ^ { 2 } + \ln x - 1 \right.$. a. Show that for all $x$ belonging to $] 0 ; + \infty \left[ , f ^ { \prime \prime } ( x ) = \frac { 1 } { x } ( 4 \ln x + 1 ) \right.$. b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$ and specify the exact value of the abscissa of the inflection point. c. Show that the curve $C _ { f }$ is above the tangent $T _ { B }$ on the interval $[ 1 ; + \infty [$.

Part C: Area calculation

1. Justify that the tangent $T _ { B }$ has the reduced equation $y = 2 x - \mathrm { e }$.
2. Using integration by parts, show that $\int _ { 1 } ^ { \mathrm { e } } x \ln x d x = \frac { \mathrm { e } ^ { 2 } + 1 } { 4 }$.
3. We denote by $\mathcal { A }$ the area of the shaded region in the figure, bounded by the curve $C _ { f }$, the tangent $T _ { B }$, and the lines with equations $x = 1$ and $x = \mathrm { e }$. We admit that $\int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { 2 } d x = \frac { \mathrm { e } ^ { 2 } - 1 } { 4 }$. Deduce the exact value of $\mathcal { A }$ in square units.

Exercise 3 (4 points)

For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.
We equip space with an orthonormal coordinate system ( $O ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).

1. We consider the points $A ( - 1 ; 0 ; 5 )$ and $B ( 3 ; 2 ; - 1 )$.

Statement 1: A parametric representation of the line ( $A B$ ) is
$$\left\{ \begin{array} { l } x = 3 - 2 t \\ y = 2 - t \\ z = - 1 + 3 t \end{array} \quad \text { with } t \in \mathbb { R } \right.$$
Statement 2: The vector $\vec { n } \left( \begin{array} { c } 5 \\ - 2 \\ 1 \end{array} \right)$ is normal to the plane $( O A B )$.
2. We consider:

• the line $d$ with parametric representation $\left\{ \begin{array} { l } x = 15 + k \\ y = 8 - k \\ z = - 6 + 2 k \end{array} \right.$ with $k \in \mathbb { R }$;
• the line $d ^ { \prime }$ with parametric representation $\left\{ \begin{array} { l } x = 1 + 4 s \\ y = 2 + 4 s \\ z = 1 - 6 s \end{array} \right.$ with $s \in \mathbb { R }$.

Statement 3: The lines $d$ and $d ^ { \prime }$ are not coplanar.
3. We consider the plane $\mathcal { P }$ with equation $x - y + z + 1 = 0$.
Statement 4: The distance from point $C ( 2 ; - 1 ; 2 )$ to the plane $\mathcal { P }$ is equal to $2 \sqrt { 3 }$.

Exercise 4 (5 points)

A team of biologists is studying the evolution of the area covered by a marine algae called seagrass, on the bottom of Alycastre Bay, near the island of Porquerolles. The studied area has a total area of 20 hectares (ha), and on July 1, 2024, seagrass covered 1 ha of this area.

Part A: study of a discrete model

For any natural integer $n$, we denote by $u _ { n }$ the area of the zone, in hectares, covered by seagrass on July 1 of the year $2024 + n$. Thus, $u _ { 0 } = 1$.
A study conducted on this area made it possible to establish that for any natural integer $n$:
$$u _ { n + 1 } = - 0,02 u _ { n } ^ { 2 } + 1,3 u _ { n }$$

1. Calculate the area that seagrass should cover on July 1, 2025 according to this model.
2. We denote by $h$ the function defined on [ 0 ; 20] by $h ( x ) = - 0,02 x ^ { 2 } + 1,3 x$. We admit that $h$ is increasing on [0;20]. a. Prove that for any natural integer $n , 1 \leq u _ { n } \leq u _ { n + 1 } \leq 20$. b. Deduce that the sequence ( $u _ { n }$ ) converges. We denote its limit by $L$. c. Justify that $\mathrm { L } = 15$.
3. The biologists wish to know after how long the area covered by seagrass will exceed 14 hectares. a. Without any calculation, justify that, according to this model, this will occur. b. Copy and complete the following algorithm so that at the end of execution, it displays the answer to the biologists' question.

\begin{verbatim} def seuil(): n=0 u=1 while ..................... : n= ............ u= ............ return n \end{verbatim}

Part B: study of a continuous model

We wish to describe the area of the studied zone covered by seagrass over time with a continuous model.
In this model, for a duration $t$, in years, elapsed from July 1, 2024, the area of the studied zone covered by seagrass is given by $f ( t )$, where $f$ is a function defined on [ $0 ; + \infty [$ satisfying:

• $f ( 0 ) = 1$;
• $f$ does not vanish on [ 0 ; $+ \infty [$;
• $f$ is differentiable on $[ 0 ; + \infty [$;
• $f$ is a solution on $\left[ 0 ; + \infty \left[ \

Part A
Below, in an orthogonal coordinate system, are the curves $\mathscr{C}_1$ and $\mathscr{C}_2$, graphical representations of two functions defined and differentiable on $\mathbb{R}$. One of the two functions represented is the derivative of the other. We will denote them $g$ and $g'$. We also specify that:

• The curve $\mathscr{C}_1$ intersects the y-axis at the point with coordinates $(0; 1)$.
• The curve $\mathscr{C}_2$ intersects the y-axis at the point with coordinates $(0; 2)$ and the x-axis at the points with coordinates $(-2; 0)$ and $(1; 0)$.

1. By justifying, associate to each of the functions $g$ and $g'$ its graphical representation.
2. Justify that the equation of the tangent line to the curve representing the function $g$ at the point with x-coordinate 0 is $y = 2x + 1$.

Part B
We consider $(E)$ the differential equation $$y + y' = (2x + 3)\mathrm{e}^{-x}$$ where $y$ is a function of the real variable $x$.

1. Show that the function $f_0$ defined for every real number $x$ by $f_0(x) = (x^2 + 3x)\mathrm{e}^{-x}$ is a particular solution of the differential equation $(E)$.
2. Solve the differential equation $(E_0): y + y' = 0$.
3. Determine the solutions of the differential equation $(E)$.
4. We admit that the function $g$ described in Part A is a solution of the differential equation $(E)$. Then determine the expression of the function $g$.
5. Determine the solutions of the differential equation $(E)$ whose curve has exactly two inflection points.

Part C
We consider the function $f$ defined for every real number $x$ by: $$f(x) = (x^2 + 3x + 2)\mathrm{e}^{-x}$$

1. Prove that the limit of the function $f$ at $+\infty$ is equal to 0.
   We also admit that the limit of the function $f$ at $-\infty$ is equal to $+\infty$.
2. We admit that the function $f$ is differentiable on $\mathbb{R}$. We denote by $f'$ the derivative function of $f$ on $\mathbb{R}$. a. Verify that, for every real number $x$, $f'(x) = (-x^2 - x + 1)\mathrm{e}^{-x}$. b. Determine the sign of the derivative function $f'$ on $\mathbb{R}$ and then deduce the variations of the function $f$ on $\mathbb{R}$.
3. Explain why the function $f$ is positive on the interval $[0; +\infty[$.
4. We will denote by $\mathscr{C}_f$ the curve representing the function $f$ in an orthogonal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. We admit that the function $F$ defined for every real number $x$ by $F(x) = (-x^2 - 5x - 7)\mathrm{e}^{-x}$ is a primitive of the function $f$. Let $\alpha$ be a positive real number. Determine the area $\mathscr{A}(\alpha)$, expressed in square units, of the region of the plane bounded by the x-axis, the curve $\mathscr{C}_f$ and the lines with equations $x = 0$ and $x = \alpha$.

5. We consider a function $h$ defined on $]0; +\infty[$ whose second derivative is defined on $]0; +\infty[$ by:
$$h''(x) = x\ln x - 3x$$
Statement 5: The function $h$ is convex on $[\mathrm{e}^3; +\infty[$.
APPENDIX Exercise 3. [Figure]