Throughout the exercise, $n$ denotes a strictly positive natural number. The purpose of the exercise is to study the equation
$$\left( E _ { n } \right) : \quad \frac { \ln ( x ) } { x } = \frac { 1 } { n }$$
with unknown strictly positive real number $x$.

Part A

Let $f$ be the function defined on the interval $] 0$; $+ \infty [$ by
$$f ( x ) = \frac { \ln ( x ) } { x }$$
It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.

1. Study the variations of function $f$.
2. Determine its maximum.

Part B

1. Show that, for $n \geqslant 3$, the equation $f ( x ) = \frac { 1 } { n }$ has a unique solution on $[ 1 ; e]$ denoted $\alpha _ { n }$.
2. From the above, for every integer $n \geqslant 3$, the real number $\alpha _ { n }$ is a solution of equation $\left( E _ { n } \right)$. a. On the graph are drawn the lines $D _ { 3 } , D _ { 4 }$ and $D _ { 5 }$ with equations respectively $y = \frac { 1 } { 3 } , y = \frac { 1 } { 4 }$, $y = \frac { 1 } { 5 }$. Conjecture the direction of variation of the sequence ( $\alpha _ { n }$ ). b. Compare, for every integer $n \geqslant 3 , f \left( \alpha _ { n } \right)$ and $f \left( \alpha _ { n + 1 } \right)$. Determine the direction of variation of the sequence $\left( \alpha _ { n } \right)$. c. Deduce that the sequence ( $\alpha _ { n }$ ) converges. It is not asked to calculate its limit.
3. It is admitted that, for every integer $n \geqslant 3$, equation $\left( E _ { n } \right)$ has another solution $\beta _ { n }$ such that $$1 \leqslant \alpha _ { n } \leqslant \mathrm { e } \leqslant \beta _ { n }$$ a. It is admitted that the sequence ( $\beta _ { n }$ ) is increasing. Establish that, for every natural number $n$ greater than or equal to 3, $$\beta _ { n } \geqslant n \frac { \beta _ { 3 } } { 3 } .$$ b. Deduce the limit of the sequence ( $\beta _ { n }$ ).

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

Part C: Area calculation

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

Exercise 3 (4 points)

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

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

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

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

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

Exercise 4 (5 points)

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

Part A: study of a discrete model

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

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

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

Part B: study of a continuous model

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

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

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

10. The graph of the function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ is shown in the figure. Then the correct conclusion is [Figure]
(A) $a > 0 , b < 0 , c > 0 , d > 0$
(B) $a > 0 , b < 0 , c < 0 , d > 0$
(C) $a < 0 , b < 0 , c < 0 , d > 0$
(D) $a > 0 , b > 0 , c > 0 , d < 0$

II. Fill in the Blank Questions

(11) $\lg \frac { 5 } { 2 } + 2 \lg 2 - \left( \frac { 1 } { 2 } \right) ^ { - 1 } =$ $\_\_\_\_$. (12) In $\triangle A B C$, $A B = \sqrt { 6 } , \angle A = 75 ^ { \circ } , \angle B = 45 ^ { \circ }$. Then $A C =$ $\_\_\_\_$. (13) In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { n } = a _ { n - 1 } + \frac { 1 } { 2 } ( n \geq 2 )$. Then the sum of the first 9 terms of the sequence $\left\{ a _ { n } \right\}$ equals $\_\_\_\_$. (14) In the rectangular coordinate system $x O y$, if the line $y = 2 a$ and the graph of the function $y = | x - a | - 1$ have only one intersection point, then the value of $a$ is $\_\_\_\_$. (15) $\triangle A B C$ is an equilateral triangle with side length 2. Given that vectors $\vec { a } , \vec { b }$ satisfy $\overrightarrow { A B } = 2 \vec { a } , \overrightarrow { A C } = 2 \vec { a } + \vec { b }$, then the correct conclusions among the following are $\_\_\_\_$. (Write out the serial numbers of all correct conclusions)
(1) $\vec { a }$ is a unit vector; (2) $\vec { b }$ is a unit vector; (3) $\vec { a } \perp \vec { b }$; (4) $\vec { b } \parallel \overrightarrow { B C }$; (5) $( 4 \vec { a } + \vec { b } ) \perp \overrightarrow { B C }$.

III. Solution Questions

4. Which of the following functions is an increasing function?
A. $f ( x ) = - x$
B. $f ( x ) = \left( \frac { 2 } { 3 } \right) ^ { x }$
C. $f ( x ) = x ^ { 2 }$
D. $f ( x ) = \sqrt [ 3 ] { x }$

(1) [4 marks] For each of the functions $f _ { 1 }$ and $f _ { 2 }$, specify the maximum domain and the zero.
$$f _ { 1 } : x \mapsto \frac { 2 x + 3 } { x ^ { 2 } - 4 } \quad f _ { 2 } : x \mapsto \ln ( x + 2 )$$
(2) [3 marks] Specify the term of a function defined on $\mathbb { R }$ whose graph has a horizontal tangent at the point (2|1), but no extremum.
(3) [5 marks] Given is the function $f$ defined on $\mathbb { R }$ with $f ( x ) = - x ^ { 3 } + 9 x ^ { 2 } - 15 x - 25$. Prove that $f$ has the following properties:
(1) The graph of $f$ has slope $-15$ at the point $x = 0$.
(2) The graph of $f$ has the $x$-axis as a tangent at the point $A ( 5 \mid f ( 5 ) )$.
(3) The tangent $t$ to the graph of the function $f$ at the point $B ( - 1 \mid f ( - 1 ) )$ can be described by the equation $y = - 36 x - 36$.
(4) [3 marks] The figure shows the graph $G _ { f }$ of a function $f$ defined on $\mathbb { R }$ with the inflection point $W ( 1 \mid 4 )$.
Using the figure, determine approximately the value of the derivative of $f$ at the point $x = 1$.
Sketch the graph of the derivative function $f ^ { \prime }$ of $f$ into the figure; in doing so, pay particular attention to the location of the zeros of $f ^ { \prime }$ and the approximate value determined for $f ^ { \prime } ( 1 )$. [Figure]
For each value of $a$ with $a \in \mathbb { R } ^ { + }$, a function $f _ { a }$ is given by $f _ { a } ( x ) = \frac { 1 } { a } \cdot x ^ { 3 } - x$ with $x \in \mathbb { R }$.
(5a) [2 marks] One of the two figures shows a graph of $f _ { a }$. Specify which figure this is. Justify your answer.
[Figure]
Fig. 1
[Figure]
Fig. 2
(5b) [3 marks] For each value of $a$, the graph of $f _ { a }$ has exactly two extrema. Determine the value of $a$ for which the graph of the function $f _ { a }$ has an extremum at the point $x = 3$.
Given is the function $f : x \mapsto 2 \cdot \left( ( \ln x ) ^ { 2 } - 1 \right)$ defined on $\mathbb { R } ^ { + }$. Figure 1 shows the graph $G _ { f }$ of $f$.
[Figure]
Fig. 1

We are given four real numbers $a \leqslant b \leqslant c \leqslant d$ such that $a + d = b + c$. Study the variations of the function $x \mapsto |x - a| - |x - b| - |x - c| + |x - d|$; show that it takes positive values. A reasoned argument supported by a graphical representation would be welcome.

Show that for all $x \in [0, 1[$
$$\frac{x^{2}}{2} + (x - 1) \ln(1 - x) \leqslant \ln(2 + x) - \ln(2 - x)$$
One may perform a function study.

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0.$$
For all $n \in \mathbb { N } ^ { * }$, show that $B _ { n }$ is a decreasing function on $\mathbb { R } ^ { + }$. One may distinguish according to whether $n$ is even or odd.

The set of all $x$ for which the function $f ( x ) = \log _ { 1 / 2 } \left( x ^ { 2 } - 2 x - 3 \right)$ is defined and monotone increasing is
(a) $( - \infty , 1 )$
(b) $( - \infty , - 1 )$
(c) $( 1 , \infty )$
(d) $( 3 , \infty )$

22. Consider the following statements in S and R : $S$ : Both $\sin x$ and $\cos x$ are decreasing functions in the interval $( \sqcap / 2 , \sqcap )$ R : If a differentiable function decreases in an interval ( $\mathrm { a } , \mathrm { b }$ ), then its derivative also decreases in (a, b). Which of the following is true :
(A) Both S and R are wrong
(B) Both S and R are correct, but R is not the correct explanation of S .
(C) S is correct and R is correct explanation for S .
(D) S is correct and R is wrong.

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only INCORRECT combination?
[A] (I) (iii) (P)
[B] (II) (iv) (Q)
[C] (III) (i) (R)
[D] (II) (iii) (P)

Let $f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \leq 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$. Then the set of all values of $b$, for which $f(x)$ has maximum value at $x = 1$, is
(1) $(-2, -1]$
(2) $[-2, -1) \cup (1, 2]$
(3) $(-2, 2)$
(4) $(-\infty, -2) \cup (2, \infty)$

14. The graph of the polynomial function
$$y = a x ^ { 5 } + b x ^ { 4 } + c x ^ { 3 } + d x ^ { 2 } + e x + f$$
is sketched, where $a , b , c , d , e$ and $f$ are real constants with $a \neq 0$.
Which one of the following is not possible?
A The graph has two local minima and two local maxima.
B The graph has one local minimum and two local maxima.
C The graph has one local minimum and one local maximum.
D The graph has no local minima or local maxima.

$f ( x ) = a x ^ { 4 } + b x ^ { 3 } + c x ^ { 2 } + d x + e$, where $a , b , c , d$, and $e$ are real numbers.
Suppose $f ( x ) = 1$ has $p$ distinct real solutions, $f ( x ) = 2$ has $q$ distinct real solutions, $f ( x ) = 3$ has $r$ distinct real solutions, and $f ( x ) = 4$ has $s$ distinct real solutions.
Which one of the following is not possible?