For each of the five following statements, indicate whether it is true or false, by justifying the answer. An unjustified answer is not taken into account. An absence of answer is not penalized.

1. We consider the function $f$ defined on the interval $]0; +\infty[$ by: $$f(x) = \ln(x) - x^2$$ Statement 1: $\lim_{x \to +\infty} f(x) = -\infty$.
   
2. We consider the differential equation $$(E): \quad -2y' + 3y = \sin x + 8\cos x$$ We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = 2\cos x - \sin x$$ Statement 2: The function $f$ is a solution of the differential equation $(E)$.
   
3. We consider the function $g$ defined on the interval $]0; +\infty[$ by: $$g(x) = \ln(3x + 1) + 8$$ We consider the sequence $(u_n)$ defined by $u_0 = 25$ and for all natural integers $n$: $$u_{n+1} = g(u_n).$$ We admit that the sequence $(u_n)$ is strictly positive. Statement 3: The sequence $(u_n)$ is decreasing.
   
4. We consider an affine function $h$ defined on $\mathbb{R}$. We denote $k$ the function defined on $\mathbb{R}$ by $k(x) = x^4 + x^2 + h(x)$. Statement 4: The function $k$ is convex on $\mathbb{R}$.
   
5. An anagram of a word is the result of a permutation of the letters of that word. Example: the word BAC has 6 anagrams: $BAC, BCA, ABC, ACB, CAB, CBA$. Statement 5: The word EULER has 120 anagrams.

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, the 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 $n$, $1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 20$. b. Deduce that the sequence $(u _ { n })$ converges. We denote by $L$ its limit. c. Justify that $L = 15$.
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 $[ 0 ; + \infty [$ of the differential equation $$\left( E _ { 1 } \right) : \quad y ^ { \prime } = 0{,}02 y ( 15 - y ) .$$

We admit that such a function $f$ exists; the purpose of this part is to determine an expression for it. We denote by $f ^ { \prime }$ the derivative function of $f$.

1. Let $g$ be the function defined on $\left[ 0 ; + \infty \left[ \text{ by } g ( t ) = \frac { 1 } { f ( t ) } \right. \right.$. Show that $g$ is a solution of the differential equation $$\left( E _ { 2 } \right) : \quad y ^ { \prime } = - 0{,}3 y + 0{,}02 .$$
2. Give the solutions of the differential equation $( E _ { 2 } )$.
3. Deduce that for all $t \in [ 0 ; + \infty [$: $$f ( t ) = \frac { 15 } { 14 \mathrm { e } ^ { - 0{,}3 t } + 1 }$$
4. Determine the limit of $f$ as $+ \infty$.
5. Solve in the interval $[ 0 ; + \infty [$ the inequality $f ( t ) > 14$. Interpret the result in the context of the exercise.

A curve $C$ has the property that the slope of the tangent at any given point $( x , y )$ on $C$ is $\frac { x ^ { 2 } + y ^ { 2 } } { 2 x y }$. a) Find the general equation for such a curve. Possible hint: let $z = \frac { y } { x }$. b) Specify all possible shapes of the curves in this family. (For example, does the family include an ellipse?)

Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be twice continuously differentiable. Suppose further that $f''(x) \geq 0$ for every $x \in \mathbb{R}$. Choose the correct statement(s) from below:
(A) $f$ is bounded;
(B) $f$ is constant;
(C) If $f$ is bounded, then it is infinitely differentiable;
(D) $\int_0^x f(t)\,\mathrm{d}t$ is infinitely differentiable with respect to $x$.

For a natural number $n$, point $\mathrm { A } _ { n }$ is a point on the $x$-axis. Point $\mathrm { A } _ { n + 1 }$ is determined according to the following rule. (가) The coordinates of point $\mathrm { A } _ { 1 }$ are $( 2,0 )$. (나) (1) Let $\mathrm { P } _ { n }$ be the point where the line passing through point $\mathrm { A } _ { n }$ and parallel to the $y$-axis meets the curve $y = \frac { 1 } { x } ( x > 0 )$.
(2) Let $\mathrm { Q } _ { n }$ be the point obtained by reflecting point $\mathrm { P } _ { n }$ about the line $y = x$.
(3) Let $\mathrm { R } _ { n }$ be the point where the line passing through point $\mathrm { Q } _ { n }$ and parallel to the $y$-axis meets the $x$-axis.
(4) Let $\mathrm { A } _ { n + 1 }$ be the point obtained by translating point $\mathrm { R } _ { n }$ by 1 unit in the direction of the $x$-axis. Let the $x$-coordinate of point $\mathrm { A } _ { n }$ be $x _ { n }$. When $x _ { 5 } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [3 points]

For a natural number $n$, point $\mathrm { A } _ { n }$ is on the $x$-axis. Point $\mathrm { A } _ { n + 1 }$ is determined according to the following rule. (가) The coordinates of point $\mathrm { A } _ { 1 }$ are $( 2,0 )$. (나) (1) Let $\mathrm { P } _ { n }$ be the point where the line passing through point $\mathrm { A } _ { n }$ and parallel to the $y$-axis meets the curve $y = \frac { 1 } { x } ( x > 0 )$.
(2) Let $\mathrm { Q } _ { n }$ be the point obtained by reflecting $\mathrm { P } _ { n }$ about the line $y = x$.
(3) Let $\mathrm { R } _ { n }$ be the point where the line passing through $\mathrm { Q } _ { n }$ and parallel to the $y$-axis meets the $x$-axis.

A function $f ( x )$ that is continuous on the entire set of real numbers satisfies the following conditions. (가) For $x \leq b$, $f ( x ) = a ( x - b ) ^ { 2 } + c$. (Here, $a$, $b$, and $c$ are constants.) (나) For all real numbers $x$, $f ( x ) = \int _ { 0 } ^ { x } \sqrt { 4 - 2 f ( t ) } \, dt$. When $\int _ { 0 } ^ { 6 } f ( x ) \, dx = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

A cubic function $f ( x )$ with positive leading coefficient satisfies the following conditions. (가) The function $f ( x )$ has a local maximum at $x = 0$ and a local minimum at $x = k$. (Here, $k$ is a constant.) (나) For all real numbers $t$ greater than 1, $\int _ { 0 } ^ { t } \left| f ^ { \prime } ( x ) \right| d x = f ( t ) + f ( 0 )$ Which of the following statements in the given options are correct? [4 points] Options ᄀ. $\int _ { 0 } ^ { k } f ^ { \prime } ( x ) d x < 0$ ㄴ. $0 < k \leq 1$ ㄷ. The local minimum value of the function $f ( x )$ is 0.
(1) ᄀ
(2) ㄷ
(3) ᄀ, ㄴ
(4) ㄴ, ㄷ
(5) ᄀ, ㄴ, ㄷ

A cubic function $f ( x )$ with leading coefficient 1 and a function $g ( x )$ that is continuous on the set of all real numbers satisfy the following conditions. Find the value of $f ( 4 )$. [4 points] (가) For all real numbers $x$, $$f ( x ) = f ( 1 ) + ( x - 1 ) f ^ { \prime } ( g ( x ) )$$ (나) The minimum value of the function $g ( x )$ is $\frac { 5 } { 2 }$. (다) $f ( 0 ) = - 3$, $f ( g ( 1 ) ) = 6$

A function $f(x)$ is continuous on the set of all real numbers, $f(x) \geq 0$ for all real numbers $x$, and $f(x) = -4xe^{4x^2}$ for $x < 0$. For all positive numbers $t$, the equation $f(x) = t$ has exactly 2 distinct real roots. Let $g(t)$ denote the smaller root and $h(t)$ denote the larger root of this equation. The two functions $g(t)$ and $h(t)$ satisfy $$2g(t) + h(t) = k \quad (k \text{ is a constant})$$ for all positive numbers $t$. If $\int_0^7 f(x)\,dx = e^4 - 1$, find the value of $\frac{f(9)}{f(8)}$. [4 points]
(1) $\frac{3}{2}e^5$
(2) $\frac{4}{3}e^7$
(3) $\frac{5}{4}e^9$
(4) $\frac{6}{5}e^{11}$
(5) $\frac{7}{6}e^{13}$

21. (This question is worth 13 points) Given $\mathrm { a } > 0$, the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } e ^ { x } \cos x$ for $\mathrm { x } \in [ 0 , + \infty )$. Let $x _ { n }$ denote the $n$-th (where $n \in \mathbb { N } ^ { * }$) extremum point of $f ( x )$ in increasing order. (I) Prove that: the sequence $\left\{ f \left( \mathrm { x } _ { \mathrm { n } } \right) \right\}$ is a geometric sequence; (II) If for all $n \in \mathbb { N } ^ { * }$, the inequality $x _ { n } \leq \left| f \left( x _ { n } \right) \right|$ always holds, find the range of $a$.

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
Show that if $f$ is a solution of $(E)$ on an interval $J$, and if $a$ is a nonzero real number, then the function $h$ defined by $h(x) = a f\left(\frac{x}{a}\right)$ is also a solution of $(E)$ on an interval that one will specify.

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
We denote $\mathscr{C}$ the image in $\mathbb{R}^2$ of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
We denote $g$ the function of a real variable with real values whose graph is $\gamma\left(\left[\frac{\pi}{4}, \pi\right]\right)$.
II.B.1) Determine the domain of definition $\Delta$ of $g$, as well as an expression for $g$. II.B.2) Verify that the restriction of $g$ to the largest open interval contained in $\Delta$ is a solution of $(E)$. II.B.3) Is this a maximal solution? If not, determine a maximal solution $m$ whose graph includes that of $g$.

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
II.C.1) Recall the statement of the existence and uniqueness theorem for maximal solutions of a nonlinear scalar differential equation subject to Cauchy conditions. II.C.2) Explain how, and possibly to what extent, this theorem applies to $(E)$. II.C.3) Are the maximal solutions given by this theorem maximal solutions of $(E)$? II.C.4) Deduce from the previous questions the maximal solutions of $(E)$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Deduce from V.D the expression of $(Lf)(x)$ for $x \in E$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
What is the value of $Lf(0)$?

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Let $f$ be fixed such that $E$ is non-empty, $x \in E$ and $a > 0$. We set $h(t) = \displaystyle\int_0^t e^{-xu} f(u)\,du$ for all $t \geqslant 0$.
VI.B.1) Show that $Lf(x+a) = a\displaystyle\int_0^{+\infty} e^{-at} h(t)\,dt$.
VI.B.2) We assume that for all $n \in \mathbb{N}$, we have $Lf(x + na) = 0$.
Show that, for all $n \in \mathbb{N}$, the integral $\displaystyle\int_0^1 u^n h\!\left(-\frac{\ln u}{a}\right)du$ converges and that it is zero.
VI.B.3) What do we deduce for the function $h$?

We assume that $f$ is positive and that $E$ is neither empty nor equal to $\mathbb{R}$. We denote by $\alpha$ its infimum.
VII.A.1) Show that if $Lf$ is bounded on $E$, then $\alpha \in E$.
VII.A.2) If $\alpha \notin E$, what can we say about $Lf(x)$ when $x$ tends to $\alpha^+$?

In this question, $f(t) = \cos t$ and $\lambda(t) = \ln(1+t)$.
VII.B.1) Determine $E$.
VII.B.2) Determine $E^{\prime}$.
VII.B.3) Show that $Lf$ admits a limit at $\alpha$, the infimum of $E$, and determine it.

For $n \in \mathbb { Z }$, we denote by $\mathcal { E } _ { n }$ the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$$ For $\alpha \in \mathbb { R }$, let $\varphi _ { \alpha }$ be the function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ defined by $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad \varphi _ { \alpha } ( t ) = t ^ { \alpha }$$
For all $n \in \mathbb { Z } ^ { * }$, determine the real numbers $\alpha$ such that $\varphi _ { \alpha }$ belongs to $\mathcal { E } _ { n }$.

For $n \in \mathbb { Z }$, we denote by $\mathcal { E } _ { n }$ the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$$
Determine $\mathcal { E } _ { n }$ for $n \in \mathbb { Z }$. We will discuss separately the case $n = 0$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$
Show that $c _ { n , f }$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad \left( c _ { n , f } \right) ^ { \prime } ( r ) = \frac { i n } { r } c _ { n , g } ( r )$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$ For $n \in \mathbb { Z }$, $\mathcal { E } _ { n }$ denotes the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$ for all $t \in \mathbb { R } _ { + } ^ { * }$.
Show that $c _ { n , f }$ belongs to $\mathcal { E } _ { n }$ and that $c _ { n , f }$ is bounded in a neighbourhood of 0. Deduce the existence of $a _ { n } \in \mathbb { C }$ such that $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = a _ { n } r ^ { | n | }$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$. We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
If $n \in \mathbb { Z }$, show that the function $\left( c _ { n , f } \right) ^ { \prime }$ is bounded on $\mathbb { R } _ { + } ^ { * }$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
Show that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are constant.