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.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another.

1. We consider the differential equation: $$(E) \quad y' = \frac{1}{2}y + 4.$$ Statement 1: The solutions of $(E)$ are the functions $f$ defined on $\mathbb{R}$ by: $$f(x) = k\mathrm{e}^{\frac{1}{2}x} - 8, \quad \text{with } k \in \mathbb{R}.$$
   
2. In a final year class, there are 18 girls and 14 boys. A volleyball team is formed by randomly choosing 3 girls and 3 boys. Statement 2: There are 297024 possibilities for forming such a team.
   
3. Let $(v_n)$ be the sequence defined for every natural integer $n$ by: $$v_n = \frac{n}{2 + \cos(n)}.$$ Statement 3: The sequence $(v_n)$ diverges to $+\infty$.
   
4. In space with respect to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $\mathrm{A}(1; 1; 2)$, $\mathrm{B}(5; -1; 8)$ and $\mathrm{C}(2; 1; 3)$. Statement 4: $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}} = 10$ and a measure of the angle $\widehat{\mathrm{BAC}}$ is $30^\circ$.
   
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 $[\mathrm{e}^3; +\infty[$.

In a laboratory, a chemical reaction is studied in a closed reactor under certain conditions. The numerical processing of experimental data made it possible to model the evolution of the temperature of this chemical reaction as a function of time. The objective of this exercise is to study this modeling. Temperature is expressed in degrees Celsius and time is expressed in minutes. Throughout the exercise, we are on the time interval $[ 0 ; 10 ]$. Parts A and B can be treated independently.

Part A

In an orthogonal coordinate system of the plane, we give below the representative curve of the temperature function as a function of time on the interval $[ 0 ; 10 ]$.

1. Determine, by graphical reading, after how much time the temperature returns to its initial value at time $t = 0$. [Figure]

We call $f$ the temperature function represented by the curve above. We specify that the function $f$ is defined and differentiable on the interval $[ 0 ; 10 ]$. It is admitted that the function $f$ can be written in the form $f ( t ) = ( a t + b ) \mathrm { e } ^ { - 0,5 t }$ where $a$ and $b$ are two real constants. 2. It is admitted that the exact value of $f ( 0 )$ is 40. Deduce the value of $b$. 3. It is admitted that $f$ satisfies the differential equation $( \mathrm { E } ) : y ^ { \prime } + 0,5 y = 60 \mathrm { e } ^ { - 0,5 t }$.
Determine the value of $a$.

Part B: Study of the function $f$

It is admitted that the function $f$ is defined for every real $t$ in the interval $[ 0 ; 10 ]$ by
$$f ( t ) = ( 60 t + 40 ) \mathrm { e } ^ { - 0,5 t }$$

1. Show that for every real $t$ in the interval $[ 0 ; 10 ]$, we have: $f ^ { \prime } ( t ) = ( 40 - 30 t ) \mathrm { e } ^ { - 0,5 t }$.
2. (a) Study the direction of variation of the function $f$ on the interval $[ 0 ; 10 ]$.

Draw up the table of variations of the function $f$ by including the images of the values present in the table. (b) Show that the equation $f ( t ) = 40$ admits a unique solution $\alpha$ strictly positive on the interval $] 0 ; 10 ]$. (c) Give an approximate value of $\alpha$ to the nearest tenth and give an interpretation of it in the context of the exercise. 3. We define the average temperature, expressed in degrees Celsius, of this chemical reaction between two times $t _ { 1 }$ and $t _ { 2 }$, expressed in minutes, by
$$\frac { 1 } { t _ { 2 } - t _ { 1 } } \int _ { t _ { 1 } } ^ { t _ { 2 } } f ( t ) d t$$
(a) Using integration by parts, show that
$$\int _ { 0 } ^ { 4 } f ( t ) d t = 320 - \frac { 800 } { \mathrm { e } ^ { 2 } }$$
(b) Deduce an approximate value, to the nearest degree Celsius, of the average temperature of this chemical reaction during the first 4 minutes.

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?)

[14 points] Let $\mathbb{R}_+$ denote the set of positive real numbers. For a continuous function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, define $A_r =$ the area bounded by the graph of $f$, X-axis, $x = 1$ and $x = r$ $B_r =$ the area bounded by the graph of $f$, X-axis, $x = r$ and $x = r^2$. Find all continuous $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ for which $A_r = B_r$ for every positive number $r$. Hints (use these or your own method): Find an equation relating $f(x)$ and $f(x^2)$. Consider the function $xf(x)$. Suppose a sequence $x_n$ converges to $b$ where all $x_n$ and $b$ are in the domain of a continuous function $g$. Then $g(x_n)$ must converge to $g(b)$. E.g., $g\left(3^{\frac{1}{n}}\right) \rightarrow g(1)$.

Find the product of all real roots of the irrational equation $x ^ { 2 } + 7 x + 10 + \sqrt { x ^ { 2 } + 7 x + 12 } = 0$. [3 points]

Find the product of all real roots of the irrational equation $\sqrt { x ^ { 2 } - 7 x + 15 } = x ^ { 2 } - 7 x + 9$. [3 points]

For the function $f ( x ) = e ^ { x + 1 } - 1$ and a natural number $n$, let the function $g ( x )$ be defined as $$g ( x ) = 100 | f ( x ) | - \sum _ { k = 1 } ^ { n } \left| f \left( x ^ { k } \right) \right|$$ Find the sum of all natural numbers $n$ such that $g ( x )$ is differentiable on the entire set of real numbers. [4 points]

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]

For a real number $k$, let $g ( x )$ be the inverse function of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x + k$. For the equation $4 f ^ { \prime } ( x ) + 12 x - 18 = \left( f ^ { \prime } \circ g \right) ( x )$ to have a real root in the closed interval $[ 0,1 ]$, let $m$ be the minimum value of $k$ and $M$ be the maximum value of $k$. Find the value of $m ^ { 2 } + M ^ { 2 }$. [4 points]

For a cubic function $f ( x )$ with leading coefficient $6 \pi$, the function $g ( x ) = \frac { 1 } { 2 + \sin ( f ( x ) ) }$ has a local maximum or minimum at $x = \alpha$, and when all $\alpha \geq 0$ are listed in increasing order as $\alpha _ { 1 }$, $\alpha _ { 2 } , \alpha _ { 3 } , \alpha _ { 4 } , \alpha _ { 5 } , \cdots$, the function $g ( x )$ satisfies the following conditions. (가) $\alpha _ { 1 } = 0$ and $g \left( \alpha _ { 1 } \right) = \frac { 2 } { 5 }$. (나) $\frac { 1 } { g \left( \alpha _ { 5 } \right) } = \frac { 1 } { g \left( \alpha _ { 2 } \right) } + \frac { 1 } { 2 }$ When $g ^ { \prime } \left( - \frac { 1 } { 2 } \right) = a \pi$, find the value of $a ^ { 2 }$. (Here, $0 < f ( 0 ) < \frac { \pi } { 2 }$.) [4 points]

Given the function $f(x) = \ln(1+x) - x + \frac{1}{2}x^2 - kx^3$, where $0 < k < \frac{1}{3}$.
(1) Prove: $f(x)$ has a unique extremum point and a unique zero point on the interval $(0, +\infty)$;
(2) Let $x_1, x_2$ be the extremum point and zero point of $f(x)$ on the interval $(0, +\infty)$ respectively.
(i) Let $g(t) = f(x_1 + t) - f(x_1 - t)$. Prove: $g(t)$ is monotonically decreasing on the interval $(0, x_1)$;
(ii) Compare the sizes of $2x_1$ and $x_2$, and prove your conclusion.

Specify the term of a function $j$ defined on $\mathbb { R }$ and invertible that satisfies the following condition: The graph of $j$ and the graph of the inverse function of $j$ have no common point.
Given is the function $f : x \mapsto 2 - \ln ( x - 1 )$ with maximal domain $D _ { f }$. The graph of $f$ is denoted by $G _ { f }$.
(1a) [3 marks] Show that $\left. D _ { f } = \right] 1 ; + \infty [$ and specify the behavior of $f$ at the boundaries of the domain.
(1b) [2 marks] Calculate the zero of $f$.
(1c) [5 marks] Describe how $G _ { f }$ is obtained step by step from the graph of the function $x \mapsto \ln x$ defined in $\mathbb { R } ^ { + }$. Use this to explain the monotonicity behavior of $G _ { f }$.
(1d) [4 marks] Show that $F : x \mapsto 3 x - ( x - 1 ) \cdot \ln ( x - 1 )$ with domain $\left. D _ { F } = \right] 1 ; + \infty [$ is an antiderivative of $f$, and determine the term of the antiderivative of $f$ that has a zero at $x = 2$.
Figure 1 shows an obstacle element in a skate park.
[Figure]
Fig. 1
The ramp of the symmetric obstacle element transitions into a horizontally running plateau, which is followed by the descent. The front and rear side surfaces run perpendicular to the horizontal ground. To describe the front side surface mathematically, a Cartesian coordinate system is chosen such that the x-axis represents the lower boundary and the y-axis represents the axis of symmetry of the surface in question. In the model, the plateau extends in the range $- 2 \leq x \leq 2$. The profile line of the descent is described for $2 \leq x \leq 8$ by the graph of the function $f$ investigated in Task 1 (see Figure 2). Here, one unit of length in the coordinate system corresponds to one meter in reality. [Figure]
(2a) [2 marks] Explain the meaning of the function value $f ( 2 )$ in the context of the problem and specify the term of the function $q$ whose graph $G _ { q }$ describes the profile line of the ramp in the model for $- 8 \leq x \leq - 2$.
(2b) [5 marks] Calculate the point $x _ { m }$ in the interval [ $2 ; 8$ ] where the local rate of change of $f$ equals the average rate of change over this interval.
(2c) [3 marks] The value $x _ { m }$ determined by calculation in Task 2b could alternatively be determined approximately without calculation using Figure 2. Explain how you would proceed.
(2d) [2 marks] Based on the model, calculate the size of the angle $\alpha$ that the plateau and the roadway enclose at the edge to the descent (see Figure 2).
(2e) [3 marks] The front side surface of the obstacle element is used as advertising space in partial areas of the ramp and descent (see Figure 1). In the model, these are two surface pieces, namely the area between $G _ { f }$ and the x-axis in the range $2 \leq x \leq 6$ and the corresponding symmetric area in the second quadrant. Using the antiderivative $F$ specified in Task 1d, calculate how many square meters are available as advertising space.
Consider the family of functions $g _ { k } : x \mapsto k x ^ { 3 } + 3 \cdot ( k + 1 ) x ^ { 2 } + 9 x$ defined on $\mathbb { R }$ with $k \in \mathbb { R } \backslash \{ 0 \}$ and the corresponding graphs $G _ { k }$. For each $k$, the graph $G _ { k }$ has exactly one inflection point $W _ { k }$.
(3a) [2 marks] Specify the behavior of $g _ { k }$ at the boundaries of the domain in dependence on $k$.
(3b) [3 marks] Determine the x-coordinate of $W _ { k }$ in dependence on $k$. (for verification: $x = - \frac { 1 } { k } - 1$ )
(3c) [4 marks] Determine the value of $k$ such that the corresponding inflection point $W _ { k }$ lies on the y-axis. Show that in this case the point $W _ { k }$ lies at the origin and the inflection tangent, i.e., the tangent to $G _ { k }$ at the point $W _ { k }$, has slope 9.
(3d) [2 marks] For the value of $k$ determined in Task 3c, Figure 3 shows the corresponding graph with its inflection tangent. In this coordinate system, the two axes have different scales. Determine the missing numerical values at the tick marks on the y-axis using an appropriate slope triangle on the inflection tangent and enter the numerical values in Figure 3. [Figure]

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 $(E)$.

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
Let $m$ be the maximal solution determined in question II.B.3).
II.D.1) Show that the solution $m$ is expandable as a power series in a neighborhood of 0. Calculate this expansion and specify its radius of convergence. II.D.2) Deduce the power series expansions of all maximal solutions of $(E)$; specify the radii of convergence of these power series.

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Show that the function $H$ defined on $\mathbb{R}$ by: $$H(x) = \int_{0}^{x} h(t) \, dt$$ is continuous, of class $\mathcal{C}^{1}$ piecewise, and periodic with period 1.

If $I$ is an interval of $\mathbb { R }$, we say that $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ satisfies (II.1) on $I$ if and only if $$\forall t \in I , \quad u ( t ) \left( u ( t ) + 2 t u ^ { \prime } ( t ) \right) = - 1$$
Let $J$ be a non-empty open interval of $\mathbb { R }$. Does there exist a polynomial function that is a solution of (II.1) on $J$?

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Let $\alpha$ be a number strictly less than $\frac{M}{2}$ and $F$ be the antiderivative of $f$ vanishing at $\alpha$. Show that for all $x$ and $y$ in $]-\infty, \frac{M}{2}[$, with $y \neq \alpha$, we have: $$f(2x) = 2\frac{F(x+y) - F(x+\alpha) - \frac{1}{4}F(2y) + \frac{1}{4}F(2\alpha)}{y - \alpha}$$

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Deduce that the function $f$ is of class $C^\infty$ on $]-\infty, M[$.

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Show that $f'' = 0$, then that the set of continuous solutions of equation (IV.1) forms an $\mathbb{R}$-vector space, for which we will determine a basis.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), and $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ is the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$.
We assume in this question that the function $\eta$ takes strictly positive values on $I \cap {]0, +\infty[}$.
1) Show that the function $f = \ln \circ \eta \circ \exp$ satisfies equation (IV.1) on an interval $]-\infty, M[$, with $M$ (possibly infinite) to be determined as a function of the interval $I$.
2) Deduce that on the interval $I \cap {]0, +\infty[}$ the function $\eta$ is of the form $$\eta : x \mapsto K_1 x^{\alpha_1}$$ with two constants $K_1 > 0$ and $\alpha_1 > 0$.
3) Show that on the interval $I \cap {]-\infty, 0[}$ the function $\eta$ is of the form $$\eta : x \mapsto K_2(-x)^{\alpha_2}$$ with two constants $K_2 < 0$ and $\alpha_2 > 0$.
4) Show that $I = \mathbb{R}$ then that the function $\eta$ is an odd function.

We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Show that any eigenvector of $\varphi_\alpha$ of degree greater than or equal to 1 vanishes at least once in the interval $]-1,1[$.

We assume $\alpha = 1$. We denote $\|\cdot\|$ the norm associated with $S_1$, and $$\varphi_1(y) : t \mapsto \left(1-t^2\right)y''(t) - 3t\,y'(t)$$ Justify that, for all $k \in \mathbb{N}$, there exists a unique polynomial eigenvector of $\varphi_1$ of degree $k$, of norm 1 and with positive leading coefficient. We denote it $T_k$.