Part B

In this part, $k$ denotes a strictly positive real number. We consider the function $f$ defined on $\mathbb { R }$ by $$f ( x ) = ( x - 1 ) \mathrm { e } ^ { - k x } + 1 .$$ We admit that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function. In the plane with an orthonormal coordinate system ( $\mathrm { O } ; \mathrm { I } , \mathrm { J }$ ), we denote $\mathscr { C } _ { f }$ the representative curve of the function $f$. The tangent line $T$ to the curve $\mathscr { C } _ { f }$ at point A with abscissa 1 intersects the ordinate axis at a point denoted B.

1. a. Prove that for all real $x$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { - k x } ( - k x + k + 1 )$$ b. Prove that the ordinate of point B is equal to $g ( k )$ where $g$ is the function defined in Part A, with $g ( x ) = 1 - \mathrm{e}^{-x}$.
2. Using Part A, prove that point B belongs to the segment [OJ].

In the Pyrenees National Park, a researcher is working on the decline of a protected species in high-mountain lakes: the ``midwife toad''. Parts I and II can be approached independently.

Part I: Effect of the introduction of a new species

In certain lakes in the Pyrenees, trout have been introduced by humans to enable fishing activities in the mountains. The researcher studied the impact of this introduction on the midwife toad population in a lake. His previous studies lead him to model the evolution of this population as a function of time by the following function $f$: $$f ( t ) = \left( 0.04 t ^ { 2 } - 8 t + 400 \right) \mathrm { e } ^ { \frac { t } { 50 } } + 40 \text { for } t \in [ 0 ; 120 ]$$ The variable $t$ represents the elapsed time, in days, from the introduction at time $t = 0$ of trout into the lake, and $f ( t )$ models the number of toads at time $t$.

1. Determine the number of toads present in the lake when the trout are introduced.
2. We admit that the function $f$ is differentiable on the interval $[0 ; 120]$ and we denote $f ^ { \prime }$ its derivative function. Show, by displaying the calculation steps, that for every real number $t$ belonging to the interval $[0 ; 120]$ we have: $$f ^ { \prime } ( t ) = t ( t - 100 ) \mathrm { e } ^ { \frac { t } { 50 } } \times 8 \times 10 ^ { - 4 }$$
3. Study the variations of the function $f$ on the interval $[0 ; 120]$, then draw up the variation table of $f$ on this interval (approximate values to the nearest hundredth will be given).
4. According to this model: a. Determine the number of days $J$ necessary for the number of toads to reach its minimum. What is this minimum number? b. Justify that, after reaching its minimum, the number of toads will one day exceed 140 individuals. c. Using a calculator, determine the duration in days from which the number of toads will exceed 140 individuals.

Part II: Effect of Chytridiomycosis on a tadpole population

One of the main causes of the decline of this toad species in high mountains is a disease, ``Chytridiomycosis'', caused by a fungus. The researcher considers that:

• Three quarters of the mountain lakes in the Pyrenees are not infected by the fungus, that is, they contain no contaminated tadpoles (toad larvae).
• In the remaining lakes, the probability that a tadpole is contaminated is 0.74.

The researcher randomly chooses a lake in the Pyrenees and takes samples from it. For the rest of the exercise, results will be rounded to the nearest thousandth when necessary. The researcher randomly takes a tadpole from the chosen lake to perform a test before releasing it. We denote $T$ the event ``The tadpole is contaminated by the disease'' and $L$ the event ``The lake is infected by the fungus''. We denote $\bar { L }$ the opposite event of $L$ and $\bar { T }$ the opposite event of $T$.

1. Copy and complete the following probability tree using the data from the problem statement.
2. Show that the probability $P ( T )$ that the sampled tadpole is contaminated is 0.185.
3. The tadpole is not contaminated. What is the probability that the lake is infected?

The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We admit that the second derivative of the function $f$ is defined on $\mathbb { R }$ by: $$f ^ { \prime \prime } ( x ) = ( 10 x + 25 ) \mathrm { e } ^ { x }$$ We can assert that: a. The function $f$ is convex on $\mathbb { R }$ b. The function $f$ is concave on $\mathbb { R }$ c. Point C is the unique inflection point of $\mathscr { C } _ { f }$ d. $\mathscr { C } _ { f }$ has no inflection point

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 2 (7 points) Themes: numerical functions and sequences This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns neither points nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
For questions 1 to 3 below, consider a function $f$ defined and twice differentiable on $\mathbb{R}$. The curve of its derivative function $f'$ is given. We admit that $f'$ has a maximum at $-\frac{3}{2}$ and that its curve intersects the x-axis at the point with coordinates $\left(-\frac{1}{2}; 0\right)$.
Question 1: a. The function $f$ has a maximum at $-\frac{3}{2}$; b. The function $f$ has a maximum at $-\frac{1}{2}$; c. The function $f$ has a minimum at $-\frac{1}{2}$; d. At the point with abscissa $-1$, the curve of function $f$ has a horizontal tangent.
Question 2: a. The function $f$ is convex on $]-\infty; -\frac{3}{2}[$; c. The curve $\mathscr{C}_f$ representing function $f$ does not have an inflection point;
Question 3: The second derivative $f''$ of function $f$ satisfies: a. $f''(x) \geqslant 0$ for $x \in ]-\infty; -\frac{1}{2}[$; b. $f''(x) \geqslant 0$ for $x \in [-2; -1]$; c. $f''\left(-\frac{3}{2}\right) = 0$; d. $f''(-3) = 0$.
Question 4: Consider three sequences $\left(u_n\right)$, $\left(v_n\right)$ and $\left(w_n\right)$. We know that, for every natural number $n$, we have: $u_n \leqslant v_n \leqslant w_n$ and furthermore: $\lim_{n \rightarrow +\infty} u_n = 1$ and $\lim_{n \rightarrow +\infty} w_n = 3$. We can then affirm that: a. the sequence $\left(v_n\right)$ converges; b. If the sequence $(u_n)$ is increasing then the sequence $(v_n)$ is bounded below by $u_0$; c. $1 \leqslant v_0 \leqslant 3$; d. the sequence $(v_n)$ diverges.
Question 5: Consider a sequence $(u_n)$ such that, for every non-zero natural number $n$: $u_n \leqslant u_{n+1} \leqslant \frac{1}{n}$. We can then affirm that: a. the sequence $(u_n)$ diverges; b. the sequence $(u_n)$ converges; c. $\lim_{n \rightarrow +\infty} u_n = 0$; d. $\lim_{n \rightarrow +\infty} u_n = 1$.
Question 6: Consider $(u_n)$ a real sequence such that for every natural number $n$, we have: $n < u_n < n+1$. We can affirm that: a. There exists a natural number $N$ such that $u_N$ is an integer; b. the sequence $(u_n)$ is increasing; c. the sequence $(u_n)$ is convergent; d. The sequence $(u_n)$ has no limit.

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 $A - B$.

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.

Exercise 4 (7 points) Themes: numerical functions, exponential function
Part A: study of two functions Consider the two functions $f$ and $g$ defined on the interval $[0; +\infty[$ by: $$f(x) = 0.06\left(-x^2 + 13.7x\right) \quad \text{and} \quad g(x) = (-0.15x + 2.2)\mathrm{e}^{0.2x} - 2.2.$$ We admit that the functions $f$ and $g$ are differentiable and we denote $f'$ and $g'$ their respective derivative functions.

1. The complete table of variations of function $f$ on the interval $[0; +\infty[$ is given.
   

   $x$	0	6.85	$+\infty$
   \multirow{2}{*}{$f(x)$}		$\nearrow f(6.85)$
   	0		$\underline{-}_{\infty}$

   
   a. Justify the limit of $f$ at $+\infty$. b. Justify the variations of function $f$. c. Solve the equation $f(x) = 0$.
2. a. Determine the limit of $g$ at $+\infty$. b. Prove that, for every real $x$ belonging to $[0; +\infty[$ we have: $g'(x) = (-0.03x + 0.29)\mathrm{e}^{0.2x}$. c. Study the variations of function $g$ and draw its table of variations on $[0; +\infty[$. Specify an approximate value to $10^{-2}$ of the maximum of $g$. d. Show that the equation $g(x) = 0$ has a unique non-zero solution and determine, to $10^{-2}$ near, an approximate value of this solution.

Part B: trajectories of a golf ball We wish to use the functions $f$ and $g$ studied in Part A to model in two different ways the trajectory of a golf ball. We assume that the terrain is perfectly flat. We will admit here that 13.7 is the value that cancels the function $f$ and an approximation of the value that cancels the function $g$. For $x$ representing the horizontal distance traveled by the ball in tens of yards after the shot (with $0 < x < 13.7$), $f(x)$ (or $g(x)$ depending on the model) represents the corresponding height of the ball above the ground, in tens of yards. The ``takeoff angle'' of the ball is called the angle between the x-axis and the tangent to the curve ($\mathscr{C}_f$ or $\mathscr{C}_g$ depending on the model) at its point with abscissa 0. A measure of the takeoff angle of the ball is a real number $d$ such that $\tan(d)$ is equal to the slope of this tangent. Similarly, the ``landing angle'' of the ball is called the angle between the x-axis and the tangent to the curve ($\mathscr{C}_f$ or $\mathscr{C}_g$ depending on the model) at its point with abscissa 13.7. A measure of the landing angle of the ball is a real number $a$ such that $\tan(a)$ is equal to the opposite of the slope of this tangent. All angles are measured in degrees.

1. First model: recall that here, the unit being tens of yards, $x$ represents the horizontal distance traveled by the ball after the shot and $f(x)$ the corresponding height of the ball. According to this model: a. What is the maximum height, in yards, reached by the ball during its trajectory? b. Verify that $f'(0) = 0.822$. c. Give a measure in degrees of the takeoff angle of the ball, rounded to the nearest tenth. (You may use the table below). d. What graphical property of the curve $\mathscr{C}_f$ allows us to justify that the takeoff and landing angles of the ball are equal?
2. Second model: recall that here, the unit being tens of yards, $x$ represents the horizontal distance traveled by the ball after the shot and $g(x)$ the corresponding height of the ball. According to this model: a. What is the maximum height, in yards, reached by the ball during its trajectory? We specify that $g'(0) = 0.29$ and $g'(13.7) \approx -1.87$. b. Give a measure in degrees of the takeoff angle of the ball, rounded to the nearest tenth. (You may use the table below). c. Justify that 62 is an approximate value, rounded to the nearest unit, of a measure in degrees of the landing angle of the ball.

Table: excerpt from a spreadsheet giving a measure in degrees of an angle when its tangent is known:

	A	B	C	D	E	F	G	H	I	J	K	L	M
1	$\tan(\theta)$	0.815	0.816	0.817	0.818	0.819	0.82	0.821	0.822	0.823	0.824	0.825	0.826
2	$\theta$ in degrees	39.18	39.21	39.25	39.28	39.32	39.35	39.39	39.42	39.45	39.49	39.52	39.56
3
4	$\tan(\theta)$	0.285	0.286	0.287	0.288	0.289	0.29	0.291	0.292	0.293	0.294	0.295	0.296
5	$\theta$ in degrees	15.91	15.96	16.01	16.07	16.12	16.17	16.23	16.28	16.33	16.38	16.44	16.49

Part C: Interrogating the models Based on a large number of observations of professional players' performances, the following average results were obtained:

Launch angle in degrees	Maximum height in yards	Landing angle in degrees	Horizontal distance in yards at the point of impact
24	32	52	137

Which model, among the two previously studied, seems most suitable for describing the ball strike by a professional player? The answer will be justified.

A biologist has modeled the evolution of a bacterial population (in thousands of entities) by the function $f$ defined on $[0; +\infty[$ by
$$f(t) = e^3 - e^{-0.5t^2 + t + 2}$$
where $t$ denotes the time in hours since the beginning of the experiment. Based on this modeling, he proposes the three statements below. For each of them, indicate, by justifying, whether it is true or false.

• Statement 1: ``The population increases permanently''.
• Statement 2: ``In the long term, the population will exceed 21000 bacteria''.
• Statement 3: ``The bacterial population will have a count of 10000 on two occasions over time''.

PART A We define on the interval $]0;+\infty[$ the function $g$ by: $$g(x) = \frac{2}{x} - \frac{1}{x^2} + \ln x \text{ where ln denotes the natural logarithm function.}$$ We admit that the function $g$ is differentiable on $]0;+\infty[ = I$ and we denote by $g'$ its derivative function.

1. Show that for $x > 0$, the sign of $g'(x)$ is that of the quadratic trinomial $(x^2 - 2x + 2)$.
2. Deduce that the function $g$ is strictly increasing on $]0;+\infty[$.
3. Show that the equation $g(x) = 0$ admits a unique solution on the interval $[0{,}5; 1]$, which we will denote $\alpha$.
4. We are given the sign table of $g$ on the interval $]0;+\infty[ = I$:
   

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

   
   Justify this sign table using the results obtained in the previous questions.

PART B We consider the function $f$ defined on the interval $]0;+\infty[ = I$ by: $$f(x) = \mathrm{e}^x \ln x.$$ We denote by $\mathscr{C}_f$ the representative curve of $f$ in an orthonormal coordinate system.

1. We admit that the function $f$ is twice differentiable on $]0;+\infty[$, we denote by $f'$ its derivative function, $f''$ its second derivative function and we admit that: for every real number $x > 0$, $f'(x) = \mathrm{e}^x\left(\frac{1}{x} + \ln x\right)$. Prove that, for every real number $x > 0$, we have: $f''(x) = \mathrm{e}^x\left(\frac{2}{x} - \frac{1}{x^2} + \ln x\right)$.
2. We may note that for every real $x > 0$, $f''(x) = \mathrm{e}^x \times g(x)$, where $g$ denotes the function studied in part A.
3. a. Draw the sign table of the function $f''$ on $]0;+\infty[$. Justify. b. Justify that the curve $\mathscr{C}_f$ admits a unique inflection point A. c. Study the convexity of the function $f$ on the interval $]0;+\infty[$. Justify.
4. a. Calculate the limits of $f$ at the boundaries of its domain of definition. b. Show that $f'(\alpha) = \frac{\mathrm{e}^\alpha}{\alpha^2}(1-\alpha)$. We recall that $\alpha$ is the unique solution of the equation $g(x) = 0$. c. Prove that $f'(\alpha) > 0$ and deduce the sign of $f'(x)$ for $x$ belonging to $]0;+\infty[$. d. Deduce the complete variation table of the function $f$ on $]0;+\infty[$.

We consider the function $f$ defined on $]0; +\infty[$ by $$f(x) = x^2 - 8\ln(x)$$ where ln denotes the natural logarithm function. We admit that $f$ is differentiable on $]0; +\infty[$, we denote by $f'$ its derivative function.

1. Determine $\lim_{x \rightarrow 0} f(x)$.
2. We admit that, for all $x > 0$, $f(x) = x^2\left(1 - 8\frac{\ln(x)}{x^2}\right)$.
   Deduce the limit: $\lim_{x \rightarrow +\infty} f(x)$.
3. Show that, for all real $x$ in $]0; +\infty[$, $f'(x) = \frac{2(x^2 - 4)}{x}$.
4. Study the variations of $f$ on $]0; +\infty[$ and draw up its complete variation table. We will specify the exact value of the minimum of $f$ on $]0; +\infty[$.
5. Prove that, on the interval $]0; 2]$, the equation $f(x) = 0$ admits a unique solution $\alpha$ (we will not seek to determine the value of $\alpha$).
6. We admit that, on the interval $[2; +\infty[$, the equation $f(x) = 0$ admits a unique solution $\beta$ (we will not seek to determine the value of $\beta$). Deduce the sign of $f$ on the interval $]0; +\infty[$.
7. For any real number $k$, we consider the function $g_k$ defined on $]0; +\infty[$ by: $$g_k(x) = x^2 - 8\ln(x) + k$$ Using the variation table of $f$, determine the smallest value of $k$ for which the function $g_k$ is positive on the interval $]0; +\infty[$.

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.

We consider the function $f$ defined on $[ 0 ; + \infty [$ by: $$f ( x ) = x \mathrm { e } ^ { - x }$$ We denote $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system of the plane. We admit that $f$ is twice differentiable on $[ 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative.

1. By noting that for all $x$ in $[ 0 ; + \infty [$, we have

$$f ( x ) = \frac { x } { \mathrm { e } ^ { x } }$$ prove that the curve $\mathscr { C } _ { f }$ has an asymptote at $+ \infty$ for which you will give an equation.
2. Prove that for all real $x$ belonging to $[ 0 ; + \infty [$ : $$f ^ { \prime } ( x ) = ( 1 - x ) \mathrm { e } ^ { - x }$$

1. Draw up the table of variations of $f$ on $[ 0 ; + \infty [$, on which you will show the values at the boundaries as well as the exact value of the extremum.
2. Determine, on the interval $[ 0 ; + \infty [$, the number of solutions of the equation

$$f ( x ) = \frac { 367 } { 1000 }$$

1. We admit that for all $x$ belonging to $[ 0 ; + \infty [$ :

$$f ^ { \prime \prime } ( x ) = \mathrm { e } ^ { - x } ( x - 2 )$$ Study the convexity of the function $f$ on the interval $[ 0 ; + \infty [$. 6. Let $a$ be a real number belonging to $[ 0 ; + \infty [$ and A the point of the curve $\mathscr { C } _ { f }$ with abscissa $a$. We denote $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at A. We denote $\mathrm { H } _ { a }$ the point of intersection of the line $T _ { a }$ and the ordinate axis. We denote $g ( a )$ the ordinate of $\mathrm { H } _ { a }$. a. Prove that a reduced equation of the tangent $T _ { a }$ is: $$y = \left[ ( 1 - a ) \mathrm { e } ^ { - a } \right] x + a ^ { 2 } \mathrm { e } ^ { - a }$$ b. Deduce the expression of $g ( a )$. c. Prove that $g ( a )$ is maximum when A is an inflection point of the curve $\mathscr { C } _ { f }$.

The curve $\mathscr{C}$ below represents a function $f$ defined and twice differentiable on $]0; +\infty[$. We know that:

• the maximum of the function $f$ is reached at the point with abscissa 3;
• the point P with abscissa 5 is the unique inflection point of the curve $\mathscr{C}$.

We have:
A. for all $x \in ]0; 5[$, $f(x)$ and $f'(x)$ have the same sign;
C. for all $x \in ]0; 5[$, $f'(x)$ and $f''(x)$ have the same sign;
B. for all $x \in ]5; +\infty[$, $f(x)$ and $f'(x)$ have the same sign;
D. for all $x \in ]5; +\infty[$, $f(x)$ and $f''(x)$ have the same sign.

We consider the function $f$ defined on the interval $]0; +\infty[$ by $$f(x) = (2 - \ln x) \times \ln x,$$ where ln denotes the natural logarithm function.
We admit that the function $f$ is twice differentiable on $]0; +\infty[$.
We denote by $C$ the representative curve of the function $f$ in an orthogonal coordinate system and $C'$ the representative curve of the function $f'$, the derivative function of the function $f$.
The curve $\boldsymbol{C}'$ is given (with its unique horizontal tangent (T)).

1. By graphical reading, with the precision that the above diagram allows, give: a. the slope of the tangent to $C$ at the point with abscissa 1. b. the largest interval on which the function $f$ is convex.
2. a. Calculate the limit of the function $f$ at $+\infty$. b. Calculate $\lim_{x \rightarrow 0} f(x)$. Interpret this result graphically.
3. Show that the curve $C$ intersects the x-axis at exactly two points, whose coordinates you will specify.
4. a. Show that for all real $x$ belonging to $]0; +\infty[$, $f'(x) = \dfrac{2(1 - \ln x)}{x}$. b. Deduce, by justifying, the table of variations of the function $f$ on $]0; +\infty[$.
5. We denote by $f''$ the second derivative of $f$ and we admit that for all real $x$ belonging to $]0; +\infty[$, $f''(x) = \dfrac{2(\ln x - 2)}{x^2}$. Determine by calculation the largest interval on which the function $f$ is convex and specify the coordinates of the inflection point of the curve $C$.

Part A Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(x + \frac{1}{2}\right)e^{-x} + x.$$

1. Determine the limits of $f$ at $-\infty$ and at $+\infty$.
2. It is admitted that $f$ is twice differentiable on $\mathbb{R}$. a. Prove that, for all $x \in \mathbb{R}$, $$f''(x) = \left(x - \frac{3}{2}\right)e^{-x}.$$ b. Deduce the variations and the minimum of the function $f'$ on $\mathbb{R}$. c. Justify that for all $x \in \mathbb{R}$, $f'(x) > 0$. d. Deduce that the equation $f(x) = 0$ admits a unique solution on $\mathbb{R}$. e. Give a value rounded to $10^{-3}$ of this solution.

Part B Consider a function $h$, defined and differentiable on $\mathbb{R}$, having an expression of the form $$h(x) = (ax + b)e^{-x} + x \text{, where } a \text{ and } b \text{ are two real numbers.}$$ In an orthonormal coordinate system shown below are:

• the representative curve $\mathscr{C}_h$ of the function $h$;
• the points A and B with coordinates respectively $(-2; -2.5)$ and $(2; 3.5)$.

1. Conjecture, with the precision allowed by the graph, the abscissae of any inflection points of the representative curve of the function $h$.
2. Knowing that the function $h$ admits on $\mathbb{R}$ a second derivative with expression $$h''(x) = -\frac{3}{2}e^{-x} + xe^{-x}.$$ validate or not the previous conjecture.
3. Determine an equation of the line (AB).
4. Knowing that the line (AB) is tangent to the representative curve of the function $h$ at the point with abscissa 0, deduce the values of $a$ and $b$.

We consider the function $f$ defined on $] 0 ; + \infty [$ by:
$$f ( x ) = 3 x + 1 - 2 x \ln ( x ) .$$
We admit that the function $f$ is twice differentiable on $] 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative. 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 and at $+ \infty$.
2. a. Prove that for every strictly positive real number $x$: $f ^ { \prime } ( x ) = 1 - 2 \ln ( x )$. b. Study the sign of $f ^ { \prime }$ and draw up the variation table of the function $f$ on the interval $] 0 ; + \infty [$. This table should include the limits as well as the exact value of the extremum.
3. a. Prove that the equation $f ( x ) = 0$ has a unique solution on $] 0 ; + \infty [$. We denote this solution by $\alpha$. b. Deduce the sign of the function $f$ on $] 0 ; + \infty [$.
4. We consider any primitive of the function $f$ on the interval $] 0$; $+ \infty [$. We denote it by $F$. Can we assert that the function $F$ is strictly decreasing on the interval $\left[ \mathrm { e } ^ { \frac { 1 } { 2 } } ; + \infty [ \right.$ ? Justify.
5. a. Study the convexity of the function $f$ on $] 0 ; + \infty [$. What is the position of the curve $\mathscr { C } _ { f }$ relative to its tangent lines? b. Determine an equation of the tangent line $T$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 1. c. Deduce from questions 5.a and 5.b that for every strictly positive real number $x$: $$\ln ( x ) \geqslant 1 - \frac { 1 } { x } .$$

Let $k$ be a strictly positive real number. The purpose of this exercise is to determine the number of solutions of the equation
$$\ln ( x ) = k x$$
with parameter $k$.
1. Graphical conjectures: Based on the graph (showing the curve $y = \ln(x)$, the line $y = x$ and the line $y = 0{,}2x$), conjecture the number of solutions of the equation $\ln ( x ) = k x$ for $k = 1$ then for $k = 0{,}2$.
2. Study of the case $k = 1$:
We consider the function $f$, defined and differentiable on $] 0 ; + \infty [$, by:
$$f ( x ) = \ln ( x ) - x .$$
We denote $f ^ { \prime }$ the derivative function of the function $f$. a. Calculate $f ^ { \prime } ( x )$. b. Study the direction of variation of the function $f$ on $] 0 ; + \infty [$.
Draw the variation table of the function $f$ showing the exact value of the extrema if there are any. The limits at the boundaries of the domain of definition are not expected. c. Deduce the number of solutions of the equation $\ln ( x ) = x$.
3. Study of the general case: $k$ is a strictly positive real number. We consider the function $g$ defined on $] 0 ; + \infty [$ by:
$$g ( x ) = \ln ( x ) - k x .$$
We admit that the variation table of the function $g$ is as follows:

$x$	0	$\frac { 1 } { k }$	$+ \infty$
$g ( x )$	$\longrightarrow$	$g \left( \frac { 1 } { k } \right)$
	$- \infty$		$- \infty$

a. Give, as a function of the sign of $g \left( \frac { 1 } { k } \right)$, the number of solutions of the equation $g ( x ) = 0$. b. Calculate $g \left( \frac { 1 } { k } \right)$ as a function of the real number $k$. c. Show that $g \left( \frac { 1 } { k } \right) > 0$ is equivalent to $\ln ( k ) < - 1$. d. Determine the set of values of $k$ for which the equation $\ln ( x ) = k x$ has exactly two solutions. e. Give, according to the values of $k$, the number of solutions of the equation $\ln ( x ) = k x$.

Exercise 3

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. The candidate will indicate on their answer sheet the number of the question and the chosen answer. No justification is required.
A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent. Throughout the exercise, $\mathbb { R }$ denotes the set of real numbers.

1. A primitive of the function $f$, defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$, is the function $F$, defined on $\mathbb { R }$, by: a. $F ( x ) = \frac { x ^ { 2 } } { 2 } \mathrm { e } ^ { x }$ b. $F ( x ) = ( x - 1 ) \mathrm { e } ^ { x }$ c. $F ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ d. $F ( x ) = x ^ { 2 } \mathrm { e } ^ { x ^ { 2 } }$
   
2. We consider the function $g$ defined by $g ( x ) = \ln \left( \frac { x - 1 } { 2 x + 4 } \right)$. The function $g$ is defined on: a. $\mathbb { R }$ b. $] - \infty ; - 2 [ \cup ] 1 ; + \infty [$ c. $] - 2 ; + \infty [$ d. $] - 2 ; 1 [$
   
3. The function $h$ defined on $\mathbb { R }$ by $h ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ is: a. concave on $\mathbb { R }$ b. convex on $\mathbb { R }$ c. convex on $] - \infty ; - 3 ]$ and concave on $[-3; + \infty [$ d. concave on $] - \infty ; - 3 ]$ and convex on $[-3; + \infty [$
   
4. A sequence ( $u _ { n }$ ) is bounded below by 3 and converges to a real number $\ell$. We can affirm that: a. $\ell = 3$ b. $\ell \geqslant 3$ c. The sequence ( $u _ { n }$ ) is decreasing. d. The sequence ( $u _ { n }$ ) is constant from a certain rank onwards.
   
5. The sequence ( $w _ { n }$ ) is defined by $w _ { 1 } = 2$ and for every strictly positive natural number $n$, $w _ { n + 1 } = \frac { 1 } { n } w _ { n }$. a. The sequence ( $w _ { n }$ ) is geometric b. The sequence ( $w _ { n }$ ) does not have a limit c. $w _ { 5 } = \frac { 1 } { 15 }$ d. The sequence ( $w _ { n }$ ) converges to 0.

The purpose of this exercise is to study the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = x \ln(x^2) - \frac{1}{x}$$
Part A: graphical readings
Below is the representative curve $(\mathscr{C}_f)$ of the function $f$, as well as the line $(T)$, tangent to the curve $(\mathscr{C}_f)$ at point A with coordinates $(1; -1)$. This tangent also passes through the point $B(0; -4)$.

1. Read graphically $f'(1)$ and give the reduced equation of the tangent $(T)$.
2. Give the intervals on which the function $f$ appears to be convex or concave. What does point A appear to represent for the curve $(\mathscr{C}_f)$?

Part B: analytical study

1. Determine, by justifying, the limit of $f$ at $+\infty$, then its limit at 0.
2. It is admitted that the function $f$ is twice differentiable on the interval $]0; +\infty[$. a. Determine $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$. b. Show that for all $x$ belonging to the interval $]0; +\infty[$, $$f''(x) = \frac{2(x+1)(x-1)}{x^3}.$$
3. a. Study the convexity of the function $f$ on the interval $]0; +\infty[$. b. Study the variations of the function $f'$, then the sign of $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$. Deduce the direction of variation of the function $f$ on the interval $]0; +\infty[$.
4. a. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on the interval $]0; +\infty[$. b. Give the value of $\alpha$ rounded to the nearest hundredth and show that $\alpha$ satisfies: $$\alpha^2 = \exp\left(\frac{1}{\alpha^2}\right)$$

Part A We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \frac{6}{1 + 5e^{-x}}$$ We have represented on the diagram below the representative curve $\mathscr{C}_f$ of the function $f$.

1. Show that point A with coordinates $(\ln 5 ; 3)$ belongs to the curve $\mathscr{C}_f$.
2. Show that the line with equation $y = 6$ is an asymptote to the curve $\mathscr{C}_f$.
3. a. We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. Show that for every real $x$, we have: $$f'(x) = \frac{30\mathrm{e}^{-x}}{\left(1 + 5\mathrm{e}^{-x}\right)^2}.$$ b. Deduce the complete table of variations of $f$ on $\mathbb{R}$.
4. We admit that:
   • $f$ is twice differentiable on $\mathbb{R}$, we denote $f''$ its second derivative;
   • for every real $x$,
   $$f''(x) = \frac{30\mathrm{e}^{-x}\left(5\mathrm{e}^{-x} - 1\right)}{\left(1 + 5\mathrm{e}^{-x}\right)^3}.$$ a. Study the convexity of $f$ on $\mathbb{R}$. In particular, we will show that the curve $\mathscr{C}_f$ admits an inflection point. b. Justify that for every real $x$ belonging to $]-\infty ; \ln 5]$, we have: $f(x) \geqslant \frac{5}{6}x + 1$.
5. We consider a function $F_k$ defined on $\mathbb{R}$ by $F_k(x) = k\ln\left(\mathrm{e}^x + 5\right)$, where $k$ is a real constant. a. Determine the value of the real $k$ so that $F_k$ is a primitive of $f$ on $\mathbb{R}$. b. Deduce that the area, in square units, of the domain bounded by the curve $\mathscr{C}_f$, the $x$-axis, the $y$-axis and the line with equation $x = \ln 5$ is equal to $6\ln\left(\frac{5}{3}\right)$.

Part B The objective of this part is to study the following differential equation: $$(E) \quad y' = y - \frac{1}{6}y^2.$$ We recall that a solution of equation $(E)$ is a function $u$ defined and differentiable on $\mathbb{R}$ such that for every real $x$, we have: $$u'(x) = u(x) - \frac{1}{6}[u(x)]^2.$$

1. Show that the function $f$ defined in part A is a solution of the differential equation $(E)$.
2. Solve the differential equation $y' = -y + \frac{1}{6}$.
3. We denote by $g$ a function differentiable on $\mathbb{R}$ that does not vanish. We denote by $h$ the function defined on $\mathbb{R}$ by $h(x) = \frac{1}{g(x)}$. We admit that $h$ is differentiable on $\mathbb{R}$. We denote $g'$ and $h'$ the derivative functions of $g$ and $h$. a. Show that if $h$ is a solution of the differential equation $y' = -y + \frac{1}{6}$, then $g$ is a solution of the differential equation $y' = y - \frac{1}{6}y^2$. b. For every positive real $m$, we consider the functions $g_m$ defined on $\mathbb{R}$ by: $$g_m(x) = \frac{6}{1 + 6m\mathrm{e}^{-x}}.$$ Show that for every positive real $m$, the function $g_m$ is a solution of the differential equation $(E): \quad y' = y - \frac{1}{6}y^2$.

Consider a function $f$ defined and twice differentiable on $]-2; +\infty[$. We denote by $\mathscr{C}_f$ its representative curve in an orthogonal coordinate system of the plane, $f'$ its derivative and $f''$ its second derivative. The curve $\mathscr{C}_f$ and its tangent $T$ at point B with abscissa $-1$ are drawn. It is specified that the line $T$ passes through the point $\mathrm{A}(0; -1)$.
Part A: exploitation of the graph.
Using the graph, answer the questions below.

1. Specify $f(-1)$ and $f'(-1)$.
2. Is the function $f$ convex on its domain of definition? Justify.
3. Conjecture the number of solutions of the equation $f(x) = 0$ and give a value rounded to $10^{-1}$ near a solution.

Part B: study of the function $f$
Consider that the function $f$ is defined on $]-2; +\infty[$ by: $$f(x) = x^2 + 2x - 1 + \ln(x+2),$$ where ln denotes the natural logarithm function.

1. Determine by calculation the limit of the function $f$ at $-2$. Interpret this result graphically.

It is admitted that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.

1. Show that for all $x > -2$, $\quad f'(x) = \frac{2x^2 + 6x + 5}{x+2}$.
2. Study the variations of the function $f$ on $]-2; +\infty[$ then draw up its complete variation table.
3. Show that the equation $f(x) = 0$ admits a unique solution $\alpha$ on $]-2; +\infty[$ and give a value of $\alpha$ rounded to $10^{-2}$ near.
4. Deduce the sign of $f(x)$ on $]-2; +\infty[$.
5. Show that $\mathscr{C}_f$ admits a unique inflection point and determine its abscissa.

Part C: a minimum distance.
Let $g$ be the function defined on $]-2; +\infty[$ by $\quad g(x) = \ln(x+2)$. We denote by $\mathscr{C}_g$ its representative curve in an orthonormal coordinate system $(O; I, J)$. Let $M$ be a point of $\mathscr{C}_g$ with abscissa $x$. The purpose of this part is to determine for which value of $x$ the distance $JM$ is minimal. Consider the function $h$ defined on $]-2; +\infty[$ by $\quad h(x) = JM^2$.

1. Justify that for all $x > -2$, we have: $\quad h(x) = x^2 + [\ln(x+2) - 1]^2$.
2. It is admitted that the function $h$ is differentiable on $]-2; +\infty[$ and we denote by $h'$ its derivative function. It is also admitted that for all real $x > -2$, $$h'(x) = \frac{2f(x)}{x+2}$$ where $f$ is the function studied in part B. a. Draw up the variation table of $h$ on $]-2; +\infty[$. The limits are not required. b. Deduce that the value of $x$ for which the distance $JM$ is minimal is $\alpha$ where $\alpha$ is the real number defined in question 4 of part B.
3. We will denote by $M_\alpha$ the point of $\mathscr{C}_g$ with abscissa $\alpha$. a. Show that $\ln(\alpha + 2) = 1 - 2\alpha - \alpha^2$. b. Deduce that the tangent to $\mathscr{C}_g$ at point $M_\alpha$ and the line $(JM_\alpha)$ are perpendicular. One may use the fact that, in an orthonormal coordinate system, two lines are perpendicular when the product of their slopes is equal to $-1$.

Part A: study of the function $\boldsymbol { f }$

The function $f$ is defined on the interval $] 0$; $+ \infty$ [ by:
$$f ( x ) = x - 2 + \frac { 1 } { 2 } \ln x$$
where ln denotes the natural logarithm function. We admit that the function $f$ is twice differentiable on $] 0 ; + \infty \left[ \right.$, we denote by $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative.

1. a. Determine, by justifying, the limits of $f$ at 0 and at $+ \infty$. b. Show that for all $x$ belonging to $] 0$; $+ \infty \left[ \right.$, we have: $f ^ { \prime } ( x ) = \frac { 2 x + 1 } { 2 x }$. c. Study the direction of variation of $f$ on $] 0 ; + \infty [$. d. Study the convexity of $f$ on $] 0 ; + \infty [$.
2. a. Show that the equation $f ( x ) = 0$ admits in $] 0 ; + \infty [$ a unique solution which we denote by $\alpha$ and justify that $\alpha$ belongs to the interval $[ 1 ; 2 ]$. b. Determine the sign of $f ( x )$ for $x \in ] 0$; $+ \infty [$. c. Show that $\ln ( \alpha ) = 2 ( 2 - \alpha )$.

Part B: study of the function $g$

The function $g$ is defined on $] 0 ; 1]$ by:
$$g ( x ) = - \frac { 7 } { 8 } x ^ { 2 } + x - \frac { 1 } { 4 } x ^ { 2 } \ln x .$$
We admit that the function $g$ is differentiable on $] 0 ; 1 ]$ and we denote by $g ^ { \prime }$ its derivative function.

1. Calculate $g ^ { \prime } ( x )$ for $\left. x \in \right] 0$; 1] then verify that $g ^ { \prime } ( x ) = x f \left( \frac { 1 } { x } \right)$.
2. a. Justify that for $x$ belonging to the interval $] 0$; $\frac { 1 } { \alpha } \left[ \right.$, we have $f \left( \frac { 1 } { x } \right) > 0$. b. We admit the following sign table:

$x$	\multicolumn{1}{|c}{0}	$\frac { 1 } { \alpha }$	1
sign of $f \left( \frac { 1 } { x } \right)$		+	0	-

Deduce the variation table of $g$ on the interval $] 0 ; 1 ]$. Images and limits are not required.

Part C: an area calculation

The following are represented on the graph below:

• The curve $\mathscr { C } _ { g }$ of the function $g$;
• The parabola $\mathscr { P }$ with equation $y = - \frac { 7 } { 8 } x ^ { 2 } + x$ on the interval $\left. ] 0 ; 1 \right]$.

We wish to calculate the area $\mathscr { A }$ of the shaded region between the curves $\mathscr { C } _ { g }$ and $\mathscr { P }$, and the lines with equations $x = \frac { 1 } { \alpha }$ and $x = 1$. We recall that $\ln ( \alpha ) = 2 ( 2 - \alpha )$.

1. a. Justify the relative position of the curves $C _ { g }$ and $\mathscr { P }$ on the interval $\left. ] 0 ; 1 \right]$. b. Prove the equality: $$\int _ { \frac { 1 } { \alpha } } ^ { 1 } x ^ { 2 } \ln x \mathrm {~d} x = \frac { - \alpha ^ { 3 } - 6 \alpha + 13 } { 9 \alpha ^ { 3 } }$$
2. Deduce the expression as a function of $\alpha$ of the area $\mathscr { A }$.

Let $f$ be the function defined on $\mathbb { R }$ by
$$f ( x ) = x \mathrm { e } ^ { - 2 x } .$$
We admit that $f$ is twice differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ the derivative of the function $f$. We denote $C _ { f }$ the representative curve of $f$ in an orthonormal coordinate system of the plane.
For each of the following statements, specify whether it is true or false, then justify the answer given.
Any answer without justification will not be taken into account.
Statement 1. For all real $x$, we have $f ^ { \prime } ( x ) = ( - 2 x + 1 ) \mathrm { e } ^ { - 2 x }$.
Statement 2. The function $f$ is a solution on $\mathbb { R }$ of the differential equation:
$$y ^ { \prime } + 2 y = \mathrm { e } ^ { - 2 x }$$
Statement 3. The function $f$ is convex on $] - \infty ; 1 ]$.
Statement 4. The equation $f ( x ) = - 1$ admits a unique solution on $\mathbb { R }$.
Statement 5. The area of the region bounded by the curve $C _ { f }$, the $x$-axis and the lines with equations $x = 0$ and $x = 1$ is equal to $\frac { 1 } { 4 } - \frac { 3 \mathrm { e } ^ { - 2 } } { 4 }$.

We consider a function $f$ defined on the interval $] 0 ; + \infty [$. We admit that it is twice differentiable on the interval $] 0 ; + \infty[$. We denote by $f ^ { \prime }$ its derivative function and $f ^ { \prime \prime }$ its second derivative function.
In an orthogonal coordinate system, we have drawn:

• the representative curve of $f$, denoted $\mathscr { C } _ { f }$ on the interval $] 0$; 3 ];
• the line $T _ { \mathrm { A } }$, tangent to $\mathscr { C } _ { f }$ at point $\mathrm { A } ( 1 ; 2 )$;
• the line $T _ { \mathrm { B } }$ tangent to $\mathscr { C } _ { f }$ at point $\mathrm { B } ( \mathrm { e } ; \mathrm { e } )$.

We further specify that the tangent $T _ { \mathrm { A } }$ passes through point $\mathrm { C } ( 3 ; 0 )$.
Part A: Graphical readings
Answer the following questions by justifying them using the graph.

1. Determine the derivative number $f ^ { \prime } ( 1 )$.
2. How many solutions does the equation $f ^ { \prime } ( x ) = 0$ have in the interval $] 0$; 3 ]?
3. What is the sign of $f ^ { \prime \prime } ( 0{,}2 )$ ?

Part B: Study of the function $f$
We admit in this part that the function $f$ is defined on the interval $] 0 ; + \infty [$ by $$f ( x ) = x \left[ 2 ( \ln x ) ^ { 2 } - 3 \ln x + 2 \right]$$ where ln denotes the natural logarithm function.

1. Solve in $\mathbb { R }$ the equation $2 X ^ { 2 } - 3 X + 2 = 0$. Deduce that $\mathscr { C } _ { f }$ does not intersect the x-axis.
2. Determine, by justifying, the limit of $f$ as $+ \infty$. We will admit that the limit of $f$ at 0 is equal to 0.
3. We admit that for all $x$ belonging to $] 0 ; + \infty [$, $f ^ { \prime } ( x ) = 2 ( \ln x ) ^ { 2 } + \ln x - 1$. a. Show that for all $x$ belonging to $] 0 ; + \infty [$, $f ^ { \prime \prime } ( x ) = \frac { 1 } { x } ( 4 \ln x + 1 )$. 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 $\mathscr { C } _ { f }$ is above the tangent $T _ { \mathrm { B } }$ on the interval $] 1 ; + \infty [$.

Part C: Area calculation

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