Consider the function $f$ defined on $[0; +\infty[$ by $$f(x) = \frac{\ln(x + 3)}{x + 3}$$

1. Show that $f$ is differentiable on $[0; +\infty[$. Study the sign of its derivative function $f'$, its possible limit at $+\infty$, and draw up the table of its variations.
2. We define the sequence $(u_n)_{n \geqslant 0}$ by its general term $u_n = \int_n^{n+1} f(x)\,\mathrm{d}x$. a. Justify that, if $n \leqslant x \leqslant n+1$, then $f(n+1) \leqslant f(x) \leqslant f(n)$. b. Show, without attempting to calculate $u_n$, that, for every natural integer $n$, $$f(n+1) \leqslant u_n \leqslant f(n).$$ c. Deduce that the sequence $(u_n)$ is convergent and determine its limit.
3. Let $F$ be the function defined on $[0; +\infty[$ by $$F(x) = [\ln(x+3)]^2.$$ a. Justify the differentiability on $[0; +\infty[$ of the function $F$ and determine, for every positive real $x$, the number $F'(x)$. b. We set, for every natural integer $n$, $I_n = \int_0^n f(x)\,\mathrm{d}x$. Calculate $I_n$.
4. We set, for every natural integer $n$, $S_n = u_0 + u_1 + \cdots + u_{n-1}$. Calculate $S_n$. Is the sequence $(S_n)$ convergent?

Let $f$ be the function defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \frac { 1 + \ln ( x ) } { x ^ { 2 } }$$
and let $\mathscr { C }$ be the representative curve of the function $f$ in a coordinate system of the plane.

1. a. Study the limit of $f$ at 0. b. What is $\lim _ { x \rightarrow + \infty } \frac { \ln ( x ) } { x }$ ? Deduce the limit of the function $f$ at $+ \infty$. c. Deduce the possible asymptotes to the curve $\mathscr { C }$.
2. a. Let $f ^ { \prime }$ denote the derivative function of the function $f$ on the interval $] 0 ; + \infty [$. Prove that, for every real $x$ belonging to the interval $] 0 ; + \infty [$, $$f ^ { \prime } ( x ) = \frac { - 1 - 2 \ln ( x ) } { x ^ { 3 } }$$ b. Solve on the interval $] 0 ; + \infty [$ the inequality $- 1 - 2 \ln ( x ) > 0$. Deduce the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$. c. Draw up the table of variations of the function $f$.
3. a. Prove that the curve $\mathscr { C }$ has a unique point of intersection with the $x$-axis, whose coordinates you will specify. b. Deduce the sign of $f ( x )$ on the interval $] 0 ; + \infty [$.
4. For every integer $n \geqslant 1$, we denote by $I _ { n }$ the area, expressed in square units, of the region bounded by the $x$-axis, the curve $\mathscr { C }$ and the lines $x = 1$ and $x = n$.

We consider the function $f$ defined for all real $x$ by $f(x) = x\mathrm{e}^{1-x^{2}}$.

1. Calculate the limit of the function $f$ at $+\infty$. Hint: you may use the fact that for all real $x$ different from 0, $f(x) = \frac{\mathrm{e}}{x} \times \frac{x^{2}}{\mathrm{e}^{x^{2}}}$. It is admitted that the limit of the function $f$ at $-\infty$ is equal to 0.
2. a. It is admitted that $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative. Prove that for all real $x$, $$f'(x) = \left(1 - 2x^{2}\right)\mathrm{e}^{1-x^{2}}$$ b. Deduce the table of variations of the function $f$.

We consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = \frac { ( \ln x ) ^ { 2 } } { x }$$
We denote $\mathscr { C }$ the representative curve of $f$ in an orthonormal coordinate system.

1. Determine the limit of the function $f$ at 0 and interpret the result graphically.
2. a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$, $$f ( x ) = 4 \left( \frac { \ln ( \sqrt { x } ) } { \sqrt { x } } \right) ^ { 2 }$$ b. Deduce that the $x$-axis is an asymptote to the representative curve of the function $f$ in the neighbourhood of $+ \infty$.
3. We admit that $f$ is differentiable on $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function. a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$, $$f ^ { \prime } ( x ) = \frac { \ln ( x ) ( 2 - \ln ( x ) ) } { x ^ { 2 } } .$$ b. Study the sign of $f ^ { \prime } ( x )$ according to the values of the strictly positive real number $x$. c. Calculate $f ( 1 )$ and $f \left( \mathrm { e } ^ { 2 } \right)$.
4. Prove that the equation $f ( x ) = 1$ admits a unique solution $\alpha$ on $] 0 ; + \infty [$ and give a bound for $\alpha$ with amplitude $10 ^ { - 2 }$.

EXERCISE-A

Main topics covered: convexity, logarithm function

Part I: graphical readings

$f$ denotes a function defined and differentiable on $\mathbb{R}$. We give below the representative curve of the derivative function $f'$.

With the precision allowed by the graph, answer the following questions

1. Determine the slope of the tangent line to the curve of function $f$ at 0.
2. a. Give the variations of the derivative function $f'$. b. Deduce an interval on which $f$ is convex.

Part II: function study

The function $f$ is defined on $\mathbb{R}$ by $$f(x) = \ln\left(x^{2} + x + \frac{5}{2}\right)$$

1. Calculate the limits of function $f$ at $+\infty$ and at $-\infty$.
2. Determine an expression $f'(x)$ of the derivative function of $f$ for all $x \in \mathbb{R}$.
3. Deduce the table of variations of $f$. Be sure to place the limits in this table.
4. a. Justify that the equation $f(x) = 2$ has a unique solution $\alpha$ in the interval $\left[-\frac{1}{2}; +\infty\right[$. b. Give an approximate value of $\alpha$ to $10^{-1}$ near.
5. The function $f'$ is differentiable on $\mathbb{R}$. We admit that, for all $x \in \mathbb{R}$, $f''(x) = \frac{-2x^{2} - 2x + 4}{\left(x^{2} + x + \frac{5}{2}\right)^{2}}$. Determine the number of inflection points of the representative curve of $f$.

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: Exponential function; differentiation; convexity

Part 1

Below is given, in the plane referred to an orthonormal reference frame, the curve representing the derivative function $f'$ of a function $f$ differentiable on $\mathbb{R}$. Using this curve, conjecture, by justifying the answers:

1. The direction of variation of the function $f$ on $\mathbb{R}$.
2. The convexity of the function $f$ on $\mathbb{R}$.

Part 2

It is admitted that the function $f$ mentioned in Part 1 is defined on $\mathbb{R}$ by: $$f(x) = (x+2)\mathrm{e}^{-x}.$$ We denote $\mathscr{C}$ the representative curve of $f$ in an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. It is admitted that the function $f$ is twice differentiable on $\mathbb{R}$, and we denote $f'$ and $f''$ the first and second derivative functions of $f$ respectively.

1. Show that, for every real number $x$, $$f(x) = \frac{x}{\mathrm{e}^x} + 2\mathrm{e}^{-x}.$$ Deduce the limit of $f$ at $+\infty$. Justify that the curve $\mathscr{C}$ admits an asymptote which you will specify. It is admitted that $\lim_{x \rightarrow -\infty} f(x) = -\infty$.
2. a. Show that, for every real number $x$, $f'(x) = (-x-1)\mathrm{e}^{-x}$. b. Study the variations on $\mathbb{R}$ of the function $f$ and draw up its variation table. c. Show that the equation $f(x) = 2$ admits a unique solution $\alpha$ on the interval $[-2;-1]$ and give an approximate value to the nearest $10^{-1}$.
3. Determine, for every real number $x$, the expression of $f''(x)$ and study the convexity of the function $f$.

What does point A with abscissa 0 represent for the curve $\mathscr{C}$?

Part I

We consider the function $h$ defined on the interval $] 0 ; + \infty [$ by: $$h ( x ) = 1 + \frac { \ln ( x ) } { x }$$

1. Determine the limit of the function $h$ at 0.
2. Determine the limit of the function $h$ at $+ \infty$.
3. We denote $h ^ { \prime }$ the derivative function of $h$. Prove that, for every real number $x$ in $] 0 ; + \infty [$, we have: $$h ^ { \prime } ( x ) = \frac { 1 - \ln ( x ) } { x ^ { 2 } }$$
4. Draw up the variation table of the function $h$ on the interval $] 0 ; + \infty [$.
5. Prove that the equation $h ( x ) = 0$ has a unique solution $\alpha$ in $] 0 ; + \infty [$. Justify that we have: $0.5 < \alpha < 0.6$.

Part II

In this part, we consider the functions $f$ and $g$ defined on $] 0 ; + \infty [$ by: $$f ( x ) = x \ln ( x ) - x ; \quad g ( x ) = \ln ( x )$$ We denote $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ the curves representing respectively the functions $f$ and $g$ in an orthonormal coordinate system $(O ; \vec { \imath } , \vec { \jmath })$. For every strictly positive real number $a$, we call:

• $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at its point with abscissa $a$;
• $D _ { a }$ the tangent to $\mathscr { C } _ { g }$ at its point with abscissa $a$.

We are looking for possible values of $a$ for which the lines $T _ { a }$ and $D _ { a }$ are perpendicular. Let $a$ be a real number belonging to the interval $] 0 ; + \infty [$.

1. Justify that the line $D _ { a }$ has slope $\frac { 1 } { a }$.
2. Justify that the line $T _ { a }$ has slope $\ln ( a )$.
3. We recall that in an orthonormal coordinate system, two lines with slopes $m$ and $m ^ { \prime }$ respectively are perpendicular if and only if $m m ^ { \prime } = - 1$. Prove that there exists a unique value of $a$, which you will identify, for which the lines $T _ { a }$ and $D _ { a }$ are perpendicular.

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: Exponential function; differentiation.
The graph below represents, in an orthogonal coordinate system, the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ defined on $\mathbb{R}$ by:
$$f(x) = x^{2}\mathrm{e}^{-x} \text{ and } g(x) = \mathrm{e}^{-x}.$$
Question 3 is independent of questions 1 and 2.

1. a. Determine the coordinates of the intersection points of $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$. b. Study the relative position of the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$.
2. For every real number $x$ in the interval $[-1; 1]$, we consider the points $M$ with coordinates $(x; f(x))$ and $N$ with coordinates $(x; g(x))$, and we denote by $d(x)$ the distance $MN$. We assume that: $d(x) = \mathrm{e}^{-x} - x^{2}\mathrm{e}^{-x}$. We assume that the function $d$ is differentiable on the interval $[-1; 1]$ and we denote by $d^{\prime}$ its derivative function. a. Show that $d^{\prime}(x) = \mathrm{e}^{-x}\left(x^{2} - 2x - 1\right)$. b. Deduce the variations of the function $d$ on the interval $[-1; 1]$. c. Determine the common abscissa $x_{0}$ of the points $M_{0}$ and $N_{0}$ allowing to obtain a maximum distance $d(x_{0})$, and give an approximate value to 0.1 of the distance $M_{0}N_{0}$.
3. Let $\Delta$ be the line with equation $y = x + 2$. We consider the function $h$ differentiable on $\mathbb{R}$ and defined by: $h(x) = \mathrm{e}^{-x} - x - 2$. By studying the number of solutions of the equation $h(x) = 0$, determine the number of intersection points of the line $\Delta$ and the curve $\mathscr{C}_{g}$.

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 2 (7 points) Theme: functions, exponential function

Part A

Let $p$ be the function defined on the interval $[ - 3 ; 4 ]$ by: $$p ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x + 1$$

1. Determine the variations of the function $p$ on the interval $[ - 3 ; 4 ]$.
2. Justify that the equation $p ( x ) = 0$ admits in the interval $[-3;4]$ a unique solution which will be denoted $\alpha$.
3. Determine an approximate value of the real number $\alpha$ to the nearest tenth.
4. Give the sign table of the function $p$ on the interval $[ - 3 ; 4 ]$.

Part B

Let $f$ be the function defined on the interval $[ - 3 ; 4 ]$ by: $$f ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + x ^ { 2 } }$$ We denote by $\mathscr { C } _ { f }$ its representative curve in an orthogonal coordinate system.

1. a. Determine the derivative of the function $f$ on the interval $[ - 3 ; 4 ]$. b. Justify that the curve $\mathscr { C } _ { f }$ admits a horizontal tangent at the point with abscissa 1.
2. The designers of a water slide use the curve $\mathscr { C } _ { f }$ as the profile of a water slide. They estimate that the water slide provides good sensations if the profile has at least two inflection points. a. Based on the graph, does the water slide seem to provide good sensations? Argue. b. It is admitted that the function $f ^ { \prime \prime }$, the second derivative of the function $f$, has the following expression for every real $x$ in the interval $[ - 3 ; 4 ]$: $$f ^ { \prime \prime } ( x ) = \frac { p ( x ) ( x - 1 ) \mathrm { e } ^ { x } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } }$$ where $p$ is the function defined in Part A. Using the above expression for $f ^ { \prime \prime }$, answer the question: ``does the water slide provide good sensations?''. Justify.

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 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 $\mathscr{C}_f$.

Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = x ^ { 2 } - 6 x + 4 \ln ( x )$$
It is admitted that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative. We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthogonal coordinate system.

1. a. Determine $\lim _ { x \rightarrow 0 } f ( x )$.
   Interpret this result graphically. b. Determine $\lim _ { x \rightarrow + \infty } f ( x )$.
2. a. Determine $f ^ { \prime } ( x )$ for all real $x$ belonging to $] 0 ; + \infty [$. b. Study the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$.
   Deduce the variation table of $f$.
3. Show that the equation $f ( x ) = 0$ has a unique solution in the interval $[4;5]$.
4. It is admitted that, for all $x$ in $] 0 ; + \infty [$, we have:
   $$f ^ { \prime \prime } ( x ) = \frac { 2 x ^ { 2 } - 4 } { x ^ { 2 } }$$
   a. Study the convexity of the function $f$ on $] 0 ; + \infty [$.
   The exact coordinates of any inflection points of $\mathscr { C } _ { f }$ will be specified. b. We denote A the point with coordinates $( \sqrt { 2 } ; f ( \sqrt { 2 } ) )$.
   Let $t$ be a strictly positive real number such that $t \neq \sqrt { 2 }$. Let $M$ be the point with coordinates $( t ; f ( t ) )$. Using question 4. a, indicate, according to the value of $t$, the relative positions of the segment [AM] and the curve $\mathscr { C } _ { f }$.

Consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = x ^ { 3 } \mathrm { e } ^ { x }$$
It is admitted that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function.

1. The sequence $(u _ { n })$ is defined by $u _ { 0 } = - 1$ and, for all natural integer $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$. a. Calculate $u _ { 1 }$ then $u _ { 2 }$.
   Exact values will be given, then approximate values to $10 ^ { - 3 }$. b. Consider the function fonc, written in Python language below.
   Recall that in Python language, ``i in range (n)'' means that i varies from 0 to n -1.
   \begin{verbatim} def fonc (n): u =- 1 for i in range(n) : u=u**3*exp(u) return u \end{verbatim}
   Determine, without justification, the value returned by fonc (2) rounded to $10 ^ { - 3 }$.
2. a. Prove that, for all real $x$, we have $f ^ { \prime } ( x ) = x ^ { 2 } \mathrm { e } ^ { x } ( x + 3 )$. b. Justify that the variation table of $f$ on $\mathbb { R }$ is the one represented below:
   

   $x$	$- \infty$	- 3	$+ \infty$
   	0		$+ \infty$
   $f$			$+ 27 \mathrm { e } ^ { - 3 }$

   
   c. Prove, by induction, that for all natural integer $n$, we have:
   $$- 1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 0$$
   d. Deduce that the sequence $(u _ { n })$ is convergent. e. We denote $\ell$ the limit of the sequence $(u _ { n })$.
   Recall that $\ell$ is a solution of the equation $f ( x ) = x$. Determine $\ell$. (For this, it will be admitted that the equation $x ^ { 2 } \mathrm { e } ^ { x } - 1 = 0$ has only one solution in $\mathbb { R }$ and that this solution is strictly greater than $\frac { 1 } { 2 }$).

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 — 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 1 (7 points) Themes: exponential function, sequences In the context of a clinical trial, two treatment protocols for a disease are being considered. The objective of this exercise is to study, for these two protocols, the evolution of the quantity of medication present in a patient's blood as a function of time.
Parts $A$ and $B$ are independent

Part A: Study of the first protocol

The first protocol consists of having the patient take a medication in tablet form. The quantity of medication present in the patient's blood, expressed in mg, is modelled by the function $f$ defined on the interval $[0; 10]$ by $$f(t) = 3t \mathrm{e}^{-0.5t + 1},$$ where $t$ denotes the time, expressed in hours, elapsed since taking the tablet.

1. a. It is admitted that the function $f$ is differentiable on the interval $[0; 10]$ and we denote $f'$ its derivative function. Show that, for every real number $t$ in $[0; 10]$, we have: $f'(t) = 3(-0.5t + 1)\mathrm{e}^{-0.5t + 1}$. b. Deduce the table of variations of the function $f$ on the interval $[0; 10]$. c. According to this model, after how much time will the quantity of medication present in the patient's blood be maximum? What is this maximum quantity?
2. a. Show that the equation $f(t) = 5$ admits a unique solution on the interval $[0; 2]$ denoted $\alpha$, of which you will give an approximate value to $10^{-2}$ near. It is admitted that the equation $f(t) = 5$ admits a unique solution on the interval $[2; 10]$, denoted $\beta$, and that an approximate value of $\beta$ to $10^{-2}$ near is 3.46. b. It is considered that this treatment is effective when the quantity of medication present in the patient's blood is greater than or equal to 5 mg. Determine, to the nearest minute, the duration of effectiveness of the medication in the case of this protocol.

Part B: Study of the second protocol

The second protocol consists of initially injecting the patient, by intravenous injection, a dose of 2 mg of medication and then re-injecting every hour a dose of $1.8$ mg. It is assumed that the medication diffuses instantaneously into the blood and is then progressively eliminated. It is estimated that when one hour has elapsed after an injection, the quantity of medication in the blood has decreased by $30\%$ compared to the quantity present immediately after this injection. This situation is modelled using the sequence $(u_n)$ where, for every natural number $n$, $u_n$ denotes the quantity of medication, expressed in mg, present in the patient's blood immediately after the injection at the $n$-th hour. We therefore have $u_0 = 2$.

1. Calculate, according to this model, the quantity $u_1$, of medication (in mg) present in the patient's blood immediately after the injection at the first hour.
2. Justify that, for every natural number $n$, we have: $u_{n+1} = 0.7u_n + 1.8$.
3. a. Show by induction that, for every natural number $n$, we have: $u_n \leqslant u_{n+1} < 6$. b. Deduce that the sequence $(u_n)$ is convergent. We denote its limit by $\ell$. c. Determine the value of $\ell$. Interpret this value in the context of the exercise.
4. Consider the sequence $(v_n)$ defined, for every natural number $n$, by $v_n = 6 - u_n$. a. Show that the sequence $(v_n)$ is a geometric sequence with ratio 0.7 and specify its first term. b. Determine the expression of $v_n$ as a function of $n$, then of $u_n$ as a function of $n$. c. With this protocol, injections are stopped when the quantity of medication present in the patient's blood is greater than or equal to $5.5$ mg. Determine, by detailing the calculations, the number of injections carried out when applying this protocol.

Exercise 3 (7 points)
Part 1
Let $g$ be the function defined for every real number $x$ in the interval $]0; +\infty[$ by: $$g(x) = \frac{2\ln x}{x}$$

1. Let $g'$ denote the derivative of $g$. Prove that for every strictly positive real $x$: $$g'(x) = \frac{2 - 2\ln x}{x^2}$$
2. We have the following variation table for the function $g$ on the interval $]0; +\infty[$:
   

   $x$	0	1	e	$+\infty$
   \begin{tabular}{ c } Variations
   of $g$

   & & & ${}^{\frac{2}{\mathrm{e}}}$ & & & & & & & & & \hline \end{tabular}
   Justify the following information read from this table: a. the value $\frac{2}{\mathrm{e}}$; b. the variations of the function $g$ on its domain; c. the limits of the function $g$ at the boundaries of its domain.
3. Deduce the sign table of the function $g$ on the interval $]0; +\infty[$.

Part 2
Let $f$ be the function defined on the interval $]0; +\infty[$ by $$f(x) = [\ln(x)]^2.$$ In this part, each study is carried out on the interval $]0; +\infty[$.

1. Prove that on the interval $]0; +\infty[$, the function $f$ is a primitive of the function $g$.
2. Using Part 1, study: a. the convexity of the function $f$; b. the variations of the function $f$.
3. a. Give an equation of the tangent line to the curve representing the function $f$ at the point with abscissa $e$. b. Deduce that, for every real $x$ in $]0; e]$: $$[\ln(x)]^2 \geqslant \frac{2}{\mathrm{e}} x - 1$$

We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \mathrm{e}^{3x} - (2x+1)\mathrm{e}^{x}$$
The purpose of this exercise is to study the function $f$ on $\mathbb{R}$.
Part A - Study of an auxiliary function
We define the function $g$ on $\mathbb{R}$ by: $$g(x) = 3\mathrm{e}^{2x} - 2x - 3$$

1. a. Determine the limit of function $g$ at $-\infty$. b. Determine the limit of function $g$ at $+\infty$.
2. a. We admit that function $g$ is differentiable on $\mathbb{R}$, and we denote by $g'$ its derivative function. Prove that for every real number $x$, we have $g'(x) = 6\mathrm{e}^{2x} - 2$. b. Study the sign of the derivative function $g'$ on $\mathbb{R}$. c. Deduce the table of variations of function $g$ on $\mathbb{R}$. Verify that function $g$ has a minimum equal to $\ln(3) - 2$.
3. a. Show that $x = 0$ is a solution of the equation $g(x) = 0$. b. Show that the equation $g(x) = 0$ has a second non-zero solution, denoted $\alpha$, for which you will give an interval of amplitude $10^{-1}$.
4. Deduce from the previous questions the sign of function $g$ on $\mathbb{R}$.

Part B - Study of function $f$

1. Function $f$ is differentiable on $\mathbb{R}$, and we denote by $f'$ its derivative function. Prove that for every real number $x$, we have $f'(x) = \mathrm{e}^{x} g(x)$, where $g$ is the function defined in Part A.
2. Deduce the sign of the derivative function $f'$ and then the variations of function $f$ on $\mathbb{R}$.
3. Why is function $f$ not convex on $\mathbb{R}$? Explain.

We admit that the function $f$ from part $\mathbf{A}$ is defined on $\mathbb{R}$ by
$$f(x) = \left(x^{2} - 5x + 6\right)\mathrm{e}^{x}$$
We denote $\mathscr{C}$ the representative curve of the function $f$ in a coordinate system.

1. a. Determine the limit of the function $f$ at $+\infty$. b. Determine the limit of the function $f$ at $-\infty$.
2. Show that, for all real $x$, we have $f^{\prime}(x) = \left(x^{2} - 3x + 1\right)\mathrm{e}^{x}$.
3. Deduce the direction of variation of the function $f$.
4. Determine the reduced equation of the tangent line $(\mathscr{T})$ to the curve $\mathscr{C}$ at the point with abscissa 0.

We admit that the function $f$ is twice differentiable on $\mathbb{R}$. We denote $f^{\prime\prime}$ the second derivative function of $f$. We admit that, for all real $x$, we have $f^{\prime\prime}(x) = (x+1)(x-2)\mathrm{e}^{x}$.
5. a. Study the convexity of the function $f$ on $\mathbb{R}$. b. Show that, for all $x$ belonging to the interval $[-1; 2]$, we have $f(x) \leqslant x + 6$.