Let $f$ be the function defined by $f ( x ) = - 2 + \ln \left( x ^ { 2 } \right)$. (a) For what real numbers $x$ is $f$ defined? (b) Find the zeros of $f$. (c) Write an equation for the line tangent to the graph of $f$ at $x = 1$.

Consider in $\mathbb{R}$ the equation: $$\ln(6x - 2) + \ln(2x - 1) = \ln(x)$$ Statement 3: the equation has two solutions in the interval $]\frac{1}{2}; +\infty[$. Indicate whether Statement 3 is true or false, justifying your answer.

We consider the function $f$ defined on $[0; +\infty[$ by $$f(x) = \ln\left(\frac{3x+1}{x+1}\right).$$ We admit that the function $f$ is differentiable on $[0; +\infty[$ and we denote by $f'$ its derivative function. We denote by $\mathscr{C}_f$ the representative curve of the function $f$ in an orthogonal coordinate system.

Part A

1. Determine $\lim_{x \rightarrow +\infty} f(x)$ and give a graphical interpretation.
2. a. Prove that, for every non-negative real number $x$, $$f'(x) = \frac{2}{(x+1)(3x+1)}$$ b. Deduce that the function $f$ is strictly increasing on $[0; +\infty[$.

Part B

Let $(u_n)$ be the sequence defined by $$u_0 = 3 \text{ and, for every natural number } n,\ u_{n+1} = f(u_n).$$

1. Prove by induction that, for every natural number $n$, $\frac{1}{2} \leqslant u_{n+1} \leqslant u_n$.
2. Prove that the sequence $(u_n)$ converges to a strictly positive limit.

Part C

We denote by $\ell$ the limit of the sequence $(u_n)$. We admit that $f(\ell) = \ell$. The objective of this part is to determine an approximate value of $\ell$. We introduce for this purpose the function $g$ defined on $[0; +\infty[$ by $g(x) = f(x) - x$. We give below the table of variations of the function $g$ on $[0; +\infty[$ where $x_0 = \frac{-2+\sqrt{7}}{3} \approx 0.215$ and $g(x_0) \approx 0.088$, rounded to $10^{-3}$.

$x$	0	$x_0$	$+\infty$
Variations		$g(x_0)$
of the
function $g$	0		$-\infty$

1. Prove that the equation $g(x) = 0$ has a unique strictly positive solution. We denote it by $\alpha$.
2. a. Copy and complete the algorithm below so that the last value taken by the variable $x$ is an approximate value of $\alpha$ by excess to 0.01 near. b. Give then the last value taken by the variable $x$ during the execution of the algorithm. $$x \leftarrow 0.22$$ While $\_\_\_\_$ do $$x \leftarrow x + 0.01$$ End While
3. Deduce an approximate value to 0.01 near of the limit $\ell$ of the sequence $(u_n)$.

Part A: establishing an inequality

On the interval $[0; +\infty[$, we define the function $f$ by $f(x) = x - \ln(x+1)$.

1. Study the monotonicity of the function $f$ on the interval $[0; +\infty[$.
2. Deduce that for all $x \in [0; +\infty[,\; \ln(x+1) \leqslant x$.

Part B: application to the study of a sequence

We set $u_0 = 1$ and for all natural number $n$, $u_{n+1} = u_n - \ln(1 + u_n)$. We admit that the sequence with general term $u_n$ is well defined.

1. Calculate an approximate value to $10^{-3}$ of $u_2$.
2. a. Prove by induction that for all natural number $n$, $u_n \geqslant 0$. b. Prove that the sequence $(u_n)$ is decreasing, and deduce that for all natural number $n$, $u_n \leqslant 1$. c. Show that the sequence $(u_n)$ is convergent.
3. Let $\ell$ denote the limit of the sequence $(u_n)$ and we admit that $\ell = f(\ell)$, where $f$ is the function defined in part A. Deduce the value of $\ell$.
4. a. Write an algorithm which, for a given natural number $p$, allows us to determine the smallest rank $N$ from which all terms of the sequence $(u_n)$ are less than $10^{-p}$. b. Determine the smallest natural number $n$ from which all terms of the sequence $(u_n)$ are less than $10^{-15}$.

Exercise 4 — Candidates who have NOT followed the specialisation course
For each of the following statements, indicate whether it is true or false, by justifying the answer.
One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. No answer is not penalised.
1. Statement 1: The equation $( 3 \ln x - 5 ) \left( e ^ { x } + 4 \right) = 0$ has exactly two real solutions.
2. Consider the sequence ( $u _ { n }$ ) defined by $$u _ { 0 } = 2 \text { and, for all natural number } n , u _ { n + 1 } = 2 u _ { n } - 5 n + 6 \text {. }$$ Statement 2: For all natural number $n , u _ { n } = 3 \times 2 ^ { n } + 5 n - 1$.
3. Consider the sequence ( $u _ { n }$ ) defined, for all natural number $n$, by $u _ { n } = n ^ { 2 } + \frac { 1 } { 2 }$.
Statement 3: The sequence $\left( u _ { n } \right)$ is geometric.
4. In a coordinate system of space, let $d$ be the line passing through point $\mathrm { A } ( - 3 ; 7 ; - 12 )$ and with direction vector $\vec { u } ( 1 ; - 2 ; 5 )$.
Let $d ^ { \prime }$ be the line with parametric representation $\left\{ \begin{array} { r l } x & = 2 t - 1 \\ y & = - 4 t + 3 \\ z & = 10 t - 2 . \end{array} , t \in \mathbf { R } \right.$
Statement 4: The lines $d$ and $d ^ { \prime }$ are coincident.
5. Consider a cube $A B C D E F G H$. The space is equipped with the orthonormal coordinate system ( $A$; $\overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E }$ ).
A parametric representation of the line (AG) is $\left\{ \begin{array} { l } x = t \\ y = t \\ z = t \end{array} \quad t \in \mathbf { R } \right.$.
Consider a point $M$ on the line (AG).
Statement 5: There are exactly two positions of point $M$ on the line (AG) such that the lines $( M \mathrm { ~B} )$ and $( M \mathrm { D } )$ are orthogonal.

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 $\ell$.

Main topics covered: Logarithm function; differentiation

Part 1

Let $h$ denote the function defined on the interval $]0; +\infty[$ by: $$h(x) = 1 + \frac{\ln(x)}{x^2}.$$ It is admitted that the function $h$ is differentiable on $]0; +\infty[$ and we denote $h'$ its derivative function.

1. Determine the limits of $h$ at 0 and at $+\infty$.
2. Show that, for every real number $x$ in $]0; +\infty[$, $h'(x) = \frac{1 - 2\ln(x)}{x^3}$.
3. Deduce the variations of the function $h$ on the interval $]0; +\infty[$.
4. Show that the equation $h(x) = 0$ admits a unique solution $\alpha$ belonging to $]0; +\infty[$ and verify that: $\frac{1}{2} < \alpha < 1$.
5. Determine the sign of $h(x)$ for $x$ belonging to $]0; +\infty[$.

Part 2

Let $f_1$ and $f_2$ denote the functions defined on $]0; +\infty[$ by: $$f_1(x) = x - 1 - \frac{\ln(x)}{x^2} \quad \text{and} \quad f_2(x) = x - 2 - \frac{2\ln(x)}{x^2}.$$ We denote $\mathscr{C}_1$ and $\mathscr{C}_2$ the respective graphs of $f_1$ and $f_2$ in a reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

1. Show that, for every real number $x$ belonging to $]0; +\infty[$, we have: $$f_1(x) - f_2(x) = h(x).$$
2. Deduce from the results of Part 1 the relative position of the curves $\mathscr{C}_1$ and $\mathscr{C}_2$.

You will justify that their unique point of intersection has coordinates $(\alpha; \alpha)$. Recall that $\alpha$ is the unique solution of the equation $h(x) = 0$.

Main topics covered: Logarithm function, limits, differentiation.

Part 1

The graph below gives the graphical representation in an orthonormal coordinate system of the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = \frac{2\ln(x) - 1}{x}$$

1. Determine by calculation the unique solution $\alpha$ of the equation $f(x) = 0$.
   Give the exact value of $\alpha$ as well as the value rounded to the nearest hundredth.
2. Specify, by graphical reading, the sign of $f(x)$ when $x$ varies in the interval $]0; +\infty[$.

Part II

We consider the function $g$ defined on the interval $]0; +\infty[$ by:
$$g(x) = [\ln(x)]^2 - \ln(x)$$

1. a. Determine the limit of the function $g$ at 0. b. Determine the limit of the function $g$ at $+\infty$.
2. We denote by $g'$ the derivative function of the function $g$ on the interval $]0; +\infty[$.
   Prove that, for any real number $x$ in $]0; +\infty[$, we have: $g'(x) = f(x)$, where $f$ denotes the function defined in Part I.
3. Draw up the variation table of the function $g$ on the interval $]0; +\infty[$.
   This table should include the limits of the function $g$ at 0 and at $+\infty$, as well as the value of the minimum of $g$ on $]0; +\infty[$.
4. Prove that, for any real number $m > -0.25$, the equation $g(x) = m$ has exactly two solutions.
5. Determine by calculation the two solutions of the equation $g(x) = 0$.

Main topics covered: Natural logarithm function; convexity
We consider the function $f$ defined on the interval $]0;+\infty[$ by: $$f(x) = x + 4 - 4\ln(x) - \frac{3}{x}$$ where ln denotes the natural logarithm function. We denote $\mathscr{C}$ the graphical representation of $f$ in an orthonormal coordinate system.

1. Determine the limit of the function $f$ at $+\infty$.
2. We assume that the function $f$ is differentiable on $]0;+\infty[$ and we denote $f^{\prime}$ its derivative function.
   Prove that, for every real number $x > 0$, we have: $$f^{\prime}(x) = \frac{x^{2} - 4x + 3}{x^{2}}$$
3. a. Give the variation table of the function $f$ on the interval $]0;+\infty[$.
   The exact values of the extrema and the limits of $f$ at 0 and at $+\infty$ will be shown. We will assume that $\lim_{x \rightarrow 0} f(x) = -\infty$. b. By simply reading the variation table, specify the number of solutions of the equation $f(x) = \frac{5}{3}$.
4. Study the convexity of the function $f$, that is, specify the parts of the interval $]0;+\infty[$ on which $f$ is convex, and those on which $f$ is concave. We will justify that the curve $\mathscr{C}$ admits a unique inflection point, whose coordinates we will specify.

Main topics covered: Logarithm function; differentiation.
Part I: Study of an auxiliary function
Let $g$ be the function defined on $]0; +\infty[$ by:
$$g(x) = \ln(x) + 2x - 2.$$

1. Determine the limits of $g$ at $+\infty$ and 0.
2. Determine the direction of variation of the function $g$ on $]0; +\infty[$.
3. Prove that the equation $g(x) = 0$ admits a unique solution $\alpha$ on $]0; +\infty[$.
4. Calculate $g(1)$ then determine the sign of $g$ on $]0; +\infty[$.

Part II: Study of a function $f$
We consider the function $f$, defined on $]0; +\infty[$ by:
$$f(x) = \left(2 - \frac{1}{x}\right)(\ln(x) - 1)$$

1. a. We assume that the function $f$ is differentiable on $]0; +\infty[$ and we denote by $f^{\prime}$ its derivative. Prove that, for every $x$ in $]0; +\infty[$, we have: $$f^{\prime}(x) = \frac{g(x)}{x^{2}}$$ b. Draw the variation table of the function $f$ on $]0; +\infty[$. The calculation of limits is not required.
2. Solve the equation $f(x) = 0$ on $]0; +\infty[$ then draw the sign table of $f$ on the interval $]0; +\infty[$.

Part III: Study of a function $F$ whose derivative is the function $f$
We assume that there exists a function $F$ differentiable on $]0; +\infty[$ whose derivative $F^{\prime}$ is the function $f$. Thus, we have: $F^{\prime} = f$. We denote by $\mathscr{C}_{F}$ the representative curve of the function $F$ in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$. We will not seek to determine an expression for $F(x)$.

1. Study the variations of $F$ on $]0; +\infty[$.
2. Does the curve $\mathscr{C}_{F}$ representative of $F$ admit tangent lines parallel to the x-axis? Justify the answer.

EXERCISE B Main topics covered: Natural logarithm function, differentiation
This exercise consists of two parts. Some results from the first part will be used in the second.

Part 1: Study of an auxiliary function

Let the function $f$ defined on the interval $[1 ; 4]$ by: $$f ( x ) = - 30 x + 50 + 35 \ln x$$

1. Recall that $f ^ { \prime }$ denotes the derivative function of the function $f$. a. For every real number $x$ in the interval $[ 1 ; 4 ]$, show that: $$f ^ { \prime } ( x ) = \frac { 35 - 30 x } { x }$$ b. Draw up the sign table of $f ^ { \prime } ( x )$ on the interval $[ 1 ; 4 ]$. c. Deduce the variations of $f$ on this same interval.
2. Justify that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $[1;4]$ then give an approximate value of $\alpha$ to $10 ^ { - 3 }$ near.
3. Draw up the sign table of $f ( x )$ for $x \in [ 1 ; 4 ]$.

Part 2: Optimisation

A company sells fruit juice. For $x$ thousand litres sold, with $x$ a real number in the interval $[ 1 ; 4 ]$, the analysis of sales leads to modelling the profit $B ( x )$ by the expression given in thousands of euros by: $$B ( x ) = - 15 x ^ { 2 } + 15 x + 35 x \ln x$$

1. According to the model, calculate the profit made by the company when it sells 2500 litres of fruit juice. Give an approximate value to the nearest euro of this profit.
2. For all $x$ in the interval $[ 1 ; 4 ]$, show that $B ^ { \prime } ( x ) = f ( x )$ where $B ^ { \prime }$ denotes the derivative function of $B$.
3. a. Using the results from part 1, give the variations of the function $B$ on the interval $[1;4]$. b. Deduce the quantity of fruit juice, to the nearest litre, that the company must sell in order to achieve maximum profit.

Exercise 1 — Multiple Choice Questionnaire (Logarithmic function)
For each of the following questions, only one of the four proposed answers is correct. The six questions are independent.

1. Consider the function $f$ defined for all real $x$ by $f(x) = \ln\left(1 + x^2\right)$.
   On $\mathbb{R}$, the equation $f(x) = 2022$ a. has no solution. b. has exactly one solution. c. has exactly two solutions. d. has infinitely many solutions.
   
2. Let the function $g$ defined for all strictly positive real $x$ by: $$g(x) = x\ln(x) - x^2$$ We denote $\mathscr{C}_g$ its representative curve in a coordinate system of the plane. a. The function $g$ is convex on $]0; +\infty[$. b. The function $g$ is concave on $]0; +\infty[$. c. The curve $\mathscr{C}_g$ has exactly one inflection point on $]0; +\infty[$. d. The curve $\mathscr{C}_g$ has exactly two inflection points on $]0; +\infty[$.
   
3. Consider the function $f$ defined on $]-1; 1[$ by $$f(x) = \frac{x}{1 - x^2}$$ An antiderivative of the function $f$ is the function $g$ defined on the interval $]-1; 1[$ by: a. $g(x) = -\frac{1}{2}\ln\left(1 - x^2\right)$ b. $g(x) = \frac{1 + x^2}{\left(1 - x^2\right)^2}$ c. $g(x) = \frac{x^2}{2\left(x - \frac{x^3}{3}\right)}$ d. $g(x) = \frac{x^2}{2}\ln\left(1 - x^2\right)$
   
4. The function $x \longmapsto \ln\left(-x^2 - x + 6\right)$ is defined on a. $]-3; 2[$ b. $]-\infty; 6]$ c. $]0; +\infty[$ d. $]2; +\infty[$
   
5. Consider the function $f$ defined on $]0.5; +\infty[$ by $$f(x) = x^2 - 4x + 3\ln(2x - 1)$$ An equation of the tangent line to the representative curve of $f$ at the point with abscissa 1 is: a. $y = 4x - 7$ b. $y = 2x - 4$ c. $y = -3(x - 1) + 4$ d. $y = 2x - 1$
   
6. The set $S$ of solutions in $\mathbb{R}$ of the inequality $\ln(x + 3) < 2\ln(x + 1)$ is: a. $S = ]-\infty; -2[ \cup ]1; +\infty[$ b. $S = ]1; +\infty[$ c. $S = \varnothing$ d. $S = ]-1; 1[$

Exercise 2: Functions, logarithm function
The purpose of this exercise is to study the function $f$, defined on $]0; +\infty[$, by: $$f(x) = 3x - x\ln(x) - 2\ln(x).$$
PART A: Study of an auxiliary function $g$
Let $g$ be the function defined on $]0; +\infty[$ by $$g(x) = 2(x-1) - x\ln(x)$$ We denote $g'$ the derivative function of $g$. We admit that $\lim_{x \rightarrow +\infty} g(x) = -\infty$.

1. Calculate $g(1)$ and $g(\mathrm{e})$.
2. Determine $\lim_{x \rightarrow 0^+} g(x)$ by justifying your approach.
3. Show that, for all $x > 0$, $g'(x) = 1 - \ln(x)$. Deduce the variation table of $g$ on $]0; +\infty[$.
4. Show that the equation $g(x) = 0$ has exactly two distinct solutions on $]0; +\infty[$: 1 and $\alpha$ with $\alpha$ belonging to the interval $[\mathrm{e}; +\infty[$. Give an approximation of $\alpha$ to 0.01.
5. Deduce the sign table of $g$ on $]0; +\infty[$.

PART B: Study of the function $f$
We consider in this part the function $f$, defined on $]0; +\infty[$, by $$f(x) = 3x - x\ln(x) - 2\ln(x).$$ We denote $f'$ the derivative function of $f$. We admit that: $\lim_{x \rightarrow 0^+} f(x) = +\infty$.

1. Determine the limit of $f$ at $+\infty$ by justifying your approach.
2. a. Justify that for all $x > 0$, $f'(x) = \dfrac{g(x)}{x}$. b. Deduce the variation table of $f$ on $]0; +\infty[$.
3. We admit that, for all $x > 0$, the second derivative of $f$, denoted $f''$, is defined by $f''(x) = \dfrac{2-x}{x^2}$. Study the convexity of $f$ and specify the coordinates of the inflection point of $\mathscr{C}_f$.

Exercise 2 — Main topics covered: functions, logarithm function.
Let $f$ be the function defined on the interval $]0; +\infty[$ by: $$f(x) = x\ln(x) - x - 2.$$ We admit that the function $f$ is twice differentiable on $]0; +\infty[$. We denote $f'$ its derivative, $f''$ its second derivative and $\mathcal{C}_f$ its representative curve in a coordinate system.

1. a. Prove that, for all $x$ belonging to $]0; +\infty[$, we have $f'(x) = \ln(x)$. b. Determine an equation of the tangent line $T$ to the curve $\mathcal{C}_f$ at the point with abscissa $x = \mathrm{e}$. c. Justify that the function $f$ is convex on the interval $]0; +\infty[$. d. Deduce the relative position of the curve $\mathcal{C}_f$ and the tangent line $T$.
   
2. a. Calculate the limit of the function $f$ at 0. b. Prove that the limit of the function $f$ at $+\infty$ is equal to $+\infty$.
   
3. Draw up the table of variations of the function $f$ on the interval $]0; +\infty[$.
   
4. a. Prove that the equation $f(x) = 0$ has a unique solution in the interval $]0; +\infty[$. We denote this solution by $\alpha$. b. Justify that the real number $\alpha$ belongs to the interval $]4{,}3; 4{,}4[$. c. Deduce the sign of the function $f$ on the interval $]0; +\infty[$.
   
5. Consider the following threshold function written in Python: Recall that the \texttt{log} function of the \texttt{math} module (which we assume is imported) denotes the natural logarithm function $\ln$. \begin{verbatim} def seuil(pas) : x=4.3 while x*log(x) - x - 2 < 0: x=x+pas return x \end{verbatim} What is the value returned when calling the function \texttt{seuil(0.01)}? Interpret this result in the context of the exercise.

Exercise 2 — 7 points Themes: Logarithm function and sequence Let $f$ be the function defined on the interval $]0;+\infty[$ by $$f(x) = x\ln(x) + 1$$ We denote $\mathscr{C}_f$ its representative curve in a coordinate system of the plane.

1. Determine the limit of the function $f$ at $0$ as well as its limit at $+\infty$.
   
2. 1. [a.] We admit that $f$ is differentiable on $]0;+\infty[$ and we denote $f'$ its derivative function. Show that for every strictly positive real number $x$: $$f'(x) = 1 + \ln(x).$$
   2. [b.] Deduce the variation table of the function $f$ on $]0;+\infty[$. The exact value of the extremum of $f$ on $]0;+\infty[$ and the limits must be shown.
   3. [c.] Justify that for all $x \in ]0;1[$, $f(x) \in ]0;1[$.
   
   
3. 1. [a.] Determine an equation of the tangent line $(T)$ to the curve $\mathscr{C}_f$ at the point with abscissa $1$.
   2. [b.] Study the convexity of the function $f$ on $]0;+\infty[$.
   3. [c.] Deduce that for every strictly positive real number $x$: $$f(x) \geqslant x$$
   
   
4. The sequence $(u_n)$ is defined by its first term $u_0$ element of the interval $]0;1[$ and for every natural number $n$: $$u_{n+1} = f(u_n)$$
   1. [a.] Prove by induction that for every natural number $n$, we have: $0 < u_n < 1$.
   2. [b.] Deduce from question 3.c. the increasing nature of the sequence $(u_n)$.
   3. [c.] Deduce that the sequence $(u_n)$ is convergent.

Exercise 3 Functions, logarithm function
Let $g$ be the function defined on the interval $]0; +\infty[$ by $$g(x) = 1 + x^{2}[1 - 2\ln(x)]$$ The function $g$ is differentiable on the interval $]0; +\infty[$ and we denote $g'$ its derivative function. We call $\mathscr{C}$ the representative curve of the function $g$ in an orthonormal coordinate system of the plane.

PART A

1. Justify that $g(\mathrm{e})$ is strictly negative.
2. Justify that $\lim_{x \rightarrow +\infty} g(x) = -\infty$.
3. a. Show that, for all $x$ belonging to the interval $]0; +\infty[$, $g'(x) = -4x\ln(x)$. b. Study the direction of variation of the function $g$ on the interval $]0; +\infty[$. c. Show that the equation $g(x) = 0$ admits a unique solution, denoted $\alpha$, on the interval $[1; +\infty[$. d. Give an interval for $\alpha$ with amplitude $10^{-2}$.
4. Deduce from the above the sign of the function $g$ on the interval $[1; +\infty[$.

PART B

1. We admit that, for all $x$ belonging to the interval $[1; \alpha]$, $g''(x) = -4[\ln(x) + 1]$. Justify that the function $g$ is concave on the interval $[1; \alpha]$.
2. In the figure opposite, A and B are points on the curve $\mathscr{C}$ with abscissae respectively 1 and $\alpha$. a. Determine the reduced equation of the line (AB). b. Deduce from this that for all real $x$ belonging to the interval $[1; \alpha]$, $$g(x) \geqslant \frac{-2}{\alpha - 1} x + \frac{2\alpha}{\alpha - 1}.$$

Exercise 3 — 7 points

Topics: Logarithm function, Sequences

Parts $\mathbf { B }$ and $\mathbf { C }$ are independent

We consider the function $f$ defined on $] 0 ; + \infty [$ by $$f ( x ) = x - x \ln x ,$$ where ln denotes the natural logarithm function.

Part A

1. Determine the limit of $f ( x )$ as $x$ tends to 0.
2. Determine the limit of $f ( x )$ as $x$ tends to $+ \infty$.
3. We admit that the function $f$ is differentiable on $] 0 ; + \infty \left[ \right.$ and we denote by $f ^ { \prime }$ its derivative function. a. Prove that, for every real number $x > 0$, we have: $f ^ { \prime } ( x ) = - \ln x$. b. Deduce the variations of the function $f$ on $] 0 ; + \infty [$ and draw its variation table.
4. Solve the equation $f ( x ) = x$ on $] 0$; $+ \infty [$.

Part B

In this part, you may use with profit certain results from Part A. We consider the sequence $(u _ { n })$ defined by: $$\begin{cases} u _ { 0 } & = 0.5 \\ u _ { n + 1 } & = u _ { n } - u _ { n } \ln u _ { n } \text { for every natural number } n , \end{cases}$$ Thus, for every natural number $n$, we have: $u _ { n + 1 } = f \left( u _ { n } \right)$.

1. We recall that the function $f$ is increasing on the interval $[ 0.5 ; 1 ]$. Prove by induction that, for every natural number $n$, we have: $0.5 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 1$.
2. a. Show that the sequence $( u _ { n } )$ is convergent. b. We denote by $\ell$ the limit of the sequence $( u _ { n } )$. Determine the value of $\ell$.

Part C

For any real number $k$, we consider the function $f _ { k }$ defined on $] 0 ; + \infty [$ by: $$f _ { k } ( x ) = k x - x \ln x$$

1. For every real number $k$, show that $f _ { k }$ admits a maximum $y _ { k }$ attained at $x _ { k } = \mathrm { e } ^ { k - 1 }$.
2. Verify that, for every real number $k$, we have: $x _ { k } = y _ { k }$.

Exercise 4 (7 points) Theme: natural logarithm function, probabilities This exercise is a multiple choice questionnaire (MCQ) comprising six questions. The six questions are independent. For each question, only one of the four answers is correct. The candidate will indicate on his answer sheet the number of the question followed by the letter corresponding to the correct answer. No justification is required.
A wrong answer, a multiple answer or no answer gives neither points nor deducts any points.
Question 1 The real number $a$ defined by $a = \ln ( 9 ) + \ln \left( \frac { \sqrt { 3 } } { 3 } \right) + \ln \left( \frac { 1 } { 9 } \right)$ is equal to: a. $1 - \frac { 1 } { 2 } \ln ( 3 )$ b. $\frac { 1 } { 2 } \ln ( 3 )$ c. $3 \ln ( 3 ) + \frac { 1 } { 2 }$ d. $- \frac { 1 } { 2 } \ln ( 3 )$
Question 2 We denote by $(E)$ the following equation $\ln x + \ln ( x - 10 ) = \ln 3 + \ln 7$ with unknown real $x$. a. 3 is a solution of $(E)$. b. $5 - \sqrt { 46 }$ is a solution of $(E)$. c. The equation $(E)$ admits a unique real solution. d. The equation $(E)$ admits two real solutions.
Question 3 The function $f$ is defined on the interval $] 0 ; + \infty [$ by the expression $f ( x ) = x ^ { 2 } ( - 1 + \ln x )$. We denote by $\mathscr { C } _ { f }$ its representative curve in the plane with a coordinate system. a. For every real $x$ in the interval $] 0 ; + \infty [$ , $f ^ { \prime } ( x ) = 2 x + \frac { 1 } { x }$. b. The function $f$ is increasing on the interval $] 0 ; + \infty [$. c. $f ^ { \prime } ( \sqrt { \mathrm { e } } )$ is different from 0. d. The line with equation $y = - \frac { 1 } { 2 } e$ is tangent to the curve $\mathscr { C } _ { f }$ at the point with abscissa $\sqrt { e }$.
Question 4
A bag contains 20 yellow tokens and 30 blue tokens. We draw successively and with replacement 5 tokens from the bag. The probability of drawing exactly 2 yellow tokens, rounded to the nearest thousandth, is: a. 0.683 b. 0.346 c. 0.230 d. 0.165
Question 5
A bag contains 20 yellow tokens and 30 blue tokens. We draw successively and with replacement 5 tokens from the bag. The probability of drawing at least one yellow token, rounded to the nearest thousandth, is: a. 0.078 b. 0.259 c. 0.337 d. 0.922
Question 6
A bag contains 20 yellow tokens and 30 blue tokens. We perform the following random experiment: we draw successively and with replacement five tokens from the bag. We note the number of yellow tokens obtained after these five draws. If we repeat this random experiment a very large number of times then, on average, the number of yellow tokens is equal to: a. 0.4 b. 1.2 c. 2 d. 2.5

Exercise 4 — Theme: Functions, Exponential Function, Logarithm Function; Sequences
Part A Consider the function $f$ defined for every real $x$ in $]0; 1]$ by: $$f(x) = \mathrm{e}^{-x} + \ln(x).$$

1. Calculate the limit of $f$ at 0.
2. It is admitted that $f$ is differentiable on $]0; 1]$. Let $f'$ denote its derivative function. Prove that, for every real $x$ belonging to $]0; 1]$, we have: $$f'(x) = \frac{1 - x\mathrm{e}^{-x}}{x}$$
3. Justify that, for every real $x$ belonging to $]0; 1]$, we have $x\mathrm{e}^{-x} < 1$. Deduce the variation table of $f$ on $]0; 1]$.
4. Prove that there exists a unique real $\ell$ belonging to $]0; 1]$ such that $f(\ell) = 0$.

Part B

1. Two sequences $(a_n)$ and $(b_n)$ are defined by: $$\left\{\begin{array}{l} a_0 = \frac{1}{10} \\ b_0 = 1 \end{array}\right. \text{ and, for every natural number } n, \left\{\begin{array}{l} a_{n+1} = \mathrm{e}^{-b_n} \\ b_{n+1} = \mathrm{e}^{-a_n} \end{array}\right.$$ a. Calculate $a_1$ and $b_1$. Approximate values to $10^{-2}$ will be given. b. Consider below the function terms, written in Python language. \begin{verbatim} def termes (n) : a=1/10 b=1 for k in range(0,n) : c= ... b = ... a = c return(a,b) \end{verbatim} Copy and complete without justification the box above so that the function termes calculates the terms of the sequences $(a_n)$ and $(b_n)$.
2. Recall that the function $x \longmapsto \mathrm{e}^{-x}$ is decreasing on $\mathbb{R}$. a. Prove by induction that, for every natural number $n$, we have: $$0 < a_n \leqslant a_{n+1} \leqslant b_{n+1} \leqslant b_n \leqslant 1$$ b. Deduce that the sequences $(a_n)$ and $(b_n)$ are convergent.
3. Let $A$ denote the limit of $(a_n)$ and $B$ denote the limit of $(b_n)$. It is admitted that $A$ and $B$ belong to the interval $]0; 1]$, and that $A = \mathrm{e}^{-B}$ and $B = \mathrm{e}^{-A}$. a. Prove that $f(A) = 0$. b. Determine $A - B$.

For every real $x$, the expression $2 + \frac { 3 \mathrm { e } ^ { - x } - 5 } { \mathrm { e } ^ { - x } + 1 }$ is equal to: a. $\frac { 5 - 3 \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$; b. $\frac { 5 + 3 \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }$; c. $\frac { 5 + 3 \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$; d. $\frac { 5 - 3 \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }$.

4. Statement 4: For every real $x > 0 , \ln ( x ) - x + 1 \leq 0$, where ln denotes the natural logarithm function.

Exercise 4 (6 points)

The purpose of this exercise is to study the stopping of a cart on a ride, from the moment it enters the braking zone at the end of the course.
We denote by $t$ the elapsed time, expressed in seconds, from the moment the cart enters the braking zone. We model the distance travelled by the cart in the braking zone, expressed in metres, as a function of $t$, using a function denoted $d$ defined on $[ 0 ; + \infty [$.
We thus have $d ( 0 ) = 0$. Furthermore, we admit that this function $d$ is differentiable on its domain of definition. We denote by $d ^ { \prime }$ its derivative function.

Part A

In the figure (Fig. 2) below, we have drawn in an orthonormal coordinate system:

• the representative curve $C _ { d }$ of the function $d$;
• the tangent $T$ to the curve $C _ { d }$ at point $A$ with abscissa 4.7;
• the asymptote $\Delta$ to $C _ { d }$ at $+ \infty$.

[Figure]
Fig. 2
In this part, no justification is expected. With the precision that the graph allows, answer the questions below. According to this model:

1. After how much time will the cart have travelled 15 m in the braking zone?
2. What minimum length must be provided for the braking zone?
3. What is the value of $d ^ { \prime } ( 4.7 )$? Interpret this result in the context of the exercise.

Part B

We recall that $t$ denotes the elapsed time, in seconds, from the moment the cart enters the braking zone. We model the instantaneous velocity of the cart, in metres per second (m.s ${ } ^ { - 1 }$ ), as a function of $t$, by a function $v$ defined on $[ 0 ; + \infty [$. We admit that:

• the function $v$ is differentiable on its domain of definition, and we denote by $v ^ { \prime }$ its derivative function;
• the function $v$ is a solution of the differential equation (E):

$$y ^ { \prime } + 0.6 y = \mathrm { e } ^ { - 0.6 t } ,$$
where $y$ is an unknown function and where $y ^ { \prime }$ is the derivative function of $y$.
We further specify that, upon arrival in the braking zone, the velocity of the cart is equal to $12 \mathrm {~m} . \mathrm { s } ^ { - 1 }$, that is $v ( 0 ) = 12$.
1. a. We consider the differential equation ( $\left. \mathrm { E } ^ { \prime } \right) : y ^ { \prime } + 0.6 y = 0$. Determine the solutions of the differential equation ( $E ^ { \prime }$ ) on [ $0 ; + \infty [$. b. Let $g$ be the function defined on $\left[ 0 ; + \infty \left[ \operatorname { by } g ( t ) = t \mathrm { e } ^ { - 0.6 t } \right. \right.$. Verify that the function $g$ is a solution of the differential equation (E). c. Deduce the solutions of the differential equation (E) on $[ 0 ; + \infty [$. d. Deduce that for every real $t$ belonging to the interval $[ 0 ; + \infty [$, we have:
$$v ( t ) = ( 12 + t ) \mathrm { e } ^ { - 0.6 t } .$$

1. In this question, we study the function $v$ on $[ 0 ; + \infty [$. a. Show that for every real $t \in \left[ 0 ; + \infty \left[ , v ^ { \prime } ( t ) = ( - 6.2 - 0.6 t ) \mathrm { e } ^ { - 0.6 t } \right. \right.$. b. By admitting that:

$$v ( t ) = 12 \mathrm { e } ^ { - 0.6 t } + \frac { 1 } { 0.6 } \times \frac { 0.6 t } { \mathrm { e } ^ { 0.6 t }

Question 170
O logaritmo de 1 000 na base 10 é
(A) 1 (B) 2 (C) 3 (D) 4 (E) 10

A equação $\log_2(x+1) = 3$ tem como solução
(A) $x = 5$ (B) $x = 6$ (C) $x = 7$ (D) $x = 8$ (E) $x = 9$

O valor de $\log_{10} 1000 + \log_{10} 0{,}01$ é
(A) $-1$ (B) $0$ (C) $1$ (D) $2$ (E) $3$

A solução da equação $e^{2x} = e^5$ é
(A) $x = 1$ (B) $x = \dfrac{5}{2}$ (C) $x = 3$ (D) $x = 4$ (E) $x = 5$