The points $( - 1 , - 1 )$ and $( 1 , - 5 )$ are on the graph of a function $y = f ( x )$ that satisfies the differential equation $\frac { d y } { d x } = x ^ { 2 } + y$. Which of the following must be true?
(A) $( 1 , - 5 )$ is a local maximum of $f$.
(B) $( 1 , - 5 )$ is a point of inflection of the graph of $f$.
(C) $( - 1 , - 1 )$ is a local maximum of $f$.
(D) $( - 1 , - 1 )$ is a local minimum of $f$.
(E) $( - 1 , - 1 )$ is a point of inflection of the graph of $f$.

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

Part A

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

Part B

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

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

Part C

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

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

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

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.

Part A

We consider the function $f$ defined on the interval $]-1; +\infty[$ by $$f(x) = 4\ln(x+1) - \frac{x^2}{25}$$ We admit that the function $f$ is differentiable on the interval $]-1; +\infty[$.

1. Determine the limit of the function $f$ at $-1$.
2. Show that, for all $x$ belonging to the interval $]-1; +\infty[$, we have: $$f'(x) = \frac{100 - 2x - 2x^2}{25(x+1)}$$
3. Study the variations of the function $f$ on the interval $]-1; +\infty[$ and then deduce that the function $f$ is strictly increasing on the interval $[2; 6.5]$.
4. We consider $h$ the function defined on the interval $[2; 6.5]$ by $h(x) = f(x) - x$.
   The table of variations of the function $h$ is given (showing $h$ increases then decreases on $[2;6.5]$ with $h(2) < 0$ and $h(6.5) < 0$ and a positive maximum in between).
   Show that the equation $h(x) = 0$ admits a unique solution $\alpha \in [2; 6.5]$.
5. Consider the following script, written in Python language: \begin{verbatim} from math import * def f(x): return 4*log(1+x)-(x**2)/25 def bornes(n) : p = 1/10**n x = 6 while f(x)-x > 0 : x = x + p return (x-p,x) \end{verbatim} We recall that in Python language:
   • the command $\log(x)$ returns the value $\ln x$;
   • the command $\mathrm{c}**\mathrm{d}$ returns the value of $c^d$.
   a. Give the values returned by the command \texttt{bornes(2)}. The values will be given rounded to the nearest hundredth. b. Interpret these values in the context of the exercise.

Part B

In this part, we may use the results obtained in Part A. We consider the sequence $(u_n)$ defined by $u_0 = 2$, and, for all natural integer $n$, $u_{n+1} = f(u_n)$.

1. Show by induction that for all natural integer $n$, $$2 \leqslant u_n \leqslant u_{n+1} < 6.5.$$
2. Deduce that the sequence $(u_n)$ converges to a limit $\ell$.
3. We recall that the real number $\alpha$, defined in Part A, is the solution of the equation $h(x) = 0$ on the interval $[2; 6.5]$. Justify that $\ell = \alpha$.

We consider the sequence $(u_n)$ defined by $u_0 = 5$ and, for all natural integers $n$: $$u_{n+1} = 2 + \ln\left(u_n^2 - 3\right)$$ We admit that this sequence is well defined.
Part A: Exploitation of Python programs

1. Copy and complete the Python script below so that \texttt{suite(k)} which takes a natural integer $k$ as parameter returns the list of the first $k$ values of the sequence $(u_n)$.
   Remark: We specify that, for any strictly positive real number $a$, $\log(a)$ returns the value of the natural logarithm of $a$.
   \begin{verbatim} def suite(k): L = [] u = 5 for i in range(......): L.append(u) u=............ return(......) \end{verbatim}
   
2. We executed \texttt{suite(9)} below. Make two conjectures: one on the direction of variation of the sequence $(u_n)$ and another on its possible convergence.
   \begin{verbatim} >>> suite(9) [ 5, 5.091042453358316, 5.131953749864703, 5.150037910978289, 5.157974010229213, 5.1614456706362954, 5.162962248594583, 5.163624356938671, 5.163913344065642] \end{verbatim}
   
3. We then created the function \texttt{mystere(n)} given below and executed \texttt{mystere(10000)}, which returned 1. Does this output contradict the conjecture made about the direction of variation of the sequence $(u_n)$? Justify.
   \begin{verbatim} def mystere(n): L = suite(n) c = 1 for i in range(n - 1): if L[i] > L[i + 1]: c = 0 return c
   >>> mystere(10000) 1 \end{verbatim}

Part B: Study of the convergence of the sequence $(u_n)$
We consider the function $g$ defined on $[2; +\infty[$ by: $$g(x) = 2 + \ln\left(x^2 - 3\right)$$ We admit that $g$ is differentiable on $[2; +\infty[$ and we denote $g'$ its derivative function.

1. Prove that the function $g$ is increasing on $[2; +\infty[$.
2. a. Prove by induction that, for all natural integers $n$: $$4 \leqslant u_n \leqslant u_{n+1} \leqslant 6$$ b. Deduce that the sequence $(u_n)$ converges.

Part C: Study of the limit value
We consider the function $f$ defined on $[2; +\infty[$ by: $$f(x) = 2 + \ln\left(x^2 - 3\right) - x$$ We admit that $f$ is differentiable on $[2; +\infty[$ and we denote $f'$ its derivative function. We give the following variation table of $f$. No justification is requested.

$x$	2	3	$+\infty$
		$\ln(6) - 1$
$f(x)$
	0		$-\infty$

1. a. Show that the equation $f(x) = 0$ has exactly two solutions on $[2; +\infty[$ which we will denote $\alpha$ and $\beta$ with $\alpha < \beta$. b. Give the exact value of $\alpha$ and an approximate value to $10^{-3}$ of $\beta$.
2. Let $\ell$ be the limit of the sequence $(u_n)$. Justify that $f(\ell) = 0$ and determine $\ell$.

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 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 $a$.

Show that if $f$ is positive, then $u$ is also positive.

We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ Show that, for all $x > 0 , | f ( x ) | \leqslant 4 C \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$.

For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by: $$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$ and $F_0$ defined by $F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$, with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$.
(a) Show that $F_\mu$ converges pointwise to $F_0$ as $\mu$ tends to 0.
(b) Show that $\phi_\mu$ converges pointwise to $\phi_0$ as $\mu$ tends to 0.

We consider the particular case for $d = 1$ given for all $y \in \mathbb{R}$ by $F(y) = 3|y|^{2/3}$ and $y_{\text{init}} = 0$. Show that this Cauchy problem admits infinitely many global solutions.

Show that $c$ is a constant solution of $(E)$, then that $(E)$ admits exactly two constant solutions denoted $c _ { 1 }$ and $c _ { 2 }$ such that $c _ { 1 } < 0 < c _ { 2 }$. Deduce the value of $c$ as a function of $c _ { 1 }$ and $c _ { 2 }$.
We admit that $y$ is decreasing on $\mathbf { R } _ { + }$ and $\lim _ { x \rightarrow + \infty } y ( x ) = c$, where $c \in \mathbf { R }$. The equation $(E)$ is: $$( E ) : \quad y ^ { \prime } ( x ) + y ( x ) + 1 = \frac { 1 } { 2 } \mathrm { e } ^ { y ( x ) }.$$

Show that if $S _ { 0 } > 0$ then the function $S$ of the solution triplet $( S , I , R )$ of $( F )$ never vanishes, and deduce that $S$ is strictly positive.
The system $(F)$ is: $$( F ) : \left\{ \begin{array} { l } S ^ { \prime } ( x ) = - I ( x ) S ( x ) \\ I ^ { \prime } ( x ) = I ( x ) S ( x ) - I ( x ) \\ R ^ { \prime } ( x ) = I ( x ) \\ S ( 0 ) = S _ { 0 } , \quad I ( 0 ) = I _ { 0 } , \quad R ( 0 ) = R _ { 0 } \end{array} \right.$$

Suppose $f ( x )$ is a twice differentiable function on $[ a , b ]$ such that $$f ( a ) = 0 = f ( b )$$ and $$x ^ { 2 } \frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } } + 4 x \frac { d f ( x ) } { d x } + 2 f ( x ) > 0 \text { for all } x \in ( a , b )$$ Then,
(A) $f$ is negative for all $x \in ( a , b )$.
(B) $f$ is positive for all $x \in ( a , b )$.
(C) $f ( x ) = 0$ for exactly one $x \in ( a , b )$.
(D) $f ( x ) = 0$ for at least two $x \in ( a , b )$.

Match the statements/expressions in Column I with the open intervals in Column II.
Column I
(A) Interval contained in the domain of definition of non-zero solutions of the differential equation $( x - 3 ) ^ { 2 } y ^ { \prime } + y = 0$
(B) Interval containing the value of the integral $$\int _ { 1 } ^ { 5 } ( x - 1 ) ( x - 2 ) ( x - 3 ) ( x - 4 ) ( x - 5 ) d x$$ (C) Interval in which at least one of the points of local maximum of $\cos ^ { 2 } x + \sin x$ lies
(D) Interval in which $\tan ^ { - 1 } ( \sin x + \cos x )$ is increasing
Column II
(p) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
(q) $\left( 0 , \frac { \pi } { 2 } \right)$
(r) $\left( \frac { \pi } { 8 } , \frac { 5 \pi } { 4 } \right)$
(s) $\left( 0 , \frac { \pi } { 8 } \right)$
(t) $( - \pi , \pi )$

Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f ( 0 ) = f ( 1 ) = 0$ and satisfies $f ^ { \prime \prime } ( x ) - 2 f ^ { \prime } ( x ) + f ( x ) \geq e ^ { x } , x \in [ 0,1 ]$.
Which of the following is true for $0 < x < 1$?
(A) $0 < f ( x ) < \infty$
(B) $- \frac { 1 } { 2 } < f ( x ) < \frac { 1 } { 2 }$
(C) $- \frac { 1 } { 4 } < f ( x ) < 1$
(D) $- \infty < f ( x ) < 0$

For $x \in \mathbb { R }$, let the function $y ( x )$ be the solution of the differential equation
$$\frac { d y } { d x } + 12 y = \cos \left( \frac { \pi } { 12 } x \right) , \quad y ( 0 ) = 0$$
Then, which of the following statements is/are TRUE ?
(A) $y ( x )$ is an increasing function
(B) $y ( x )$ is a decreasing function
(C) There exists a real number $\beta$ such that the line $y = \beta$ intersects the curve $y = y ( x )$ at infinitely many points
(D) $y ( x )$ is a periodic function

Let $S = \left\{ ( \lambda , \mu ) \in R \times R : f ( t ) = \left( | \lambda | e ^ { | t | } - \mu \right) \sin ( 2 | t | ) , t \in R \right.$ is a differentiable function $\}$. Then, $S$ is a subset of :
(1) $( - \infty , 0 ) \times R$
(2) $R \times [ 0 , \infty )$
(3) $[ 0 , \infty ) \times R$
(4) $R \times ( - \infty , 0 )$

Let $S = \left\{ ( \lambda , \mu ) \in R \times R : f ( t ) = \left( | \lambda | e ^ { t } - \mu \right) \cdot \sin ( 2 | t | ) , t \in R \right.$, is a differentiable function $\}$. Then $S$ is a subest of?
(1) $R \times [ 0 , \infty )$
(2) $( - \infty , 0 ) \times R$
(3) $[ 0 , \infty ) \times R$
(4) $R \times ( - \infty , 0 )$

Let $y = y _ { 1 } ( x )$ and $y = y _ { 2 } ( x )$ be two distinct solutions of the differential equation $\frac { d y } { d x } = x + y$, with $y _ { 1 } ( 0 ) = 0$ and $y _ { 2 } ( 0 ) = 1$ respectively. Then, the number of points of intersection of $y = y _ { 1 } ( x )$ and $y = y _ { 2 } ( x )$ is
(1) 0
(2) 1
(3) 2
(4) 3