This exercise is a multiple choice questionnaire (MCQ). For each question, three statements are proposed, only one of these statements is correct.

1. Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(x^2 - 2x - 1\right)\mathrm{e}^x$$ A. The derivative function of $f$ is the function defined by $f^{\prime}(x) = (2x-2)\mathrm{e}^x$.
   B. The function $f$ is decreasing on the interval $]-\infty; 2]$.
   C. $\lim_{x \rightarrow -\infty} f(x) = 0$.
   
2. Consider the function $f$ defined on $\mathbb{R}$ by $f(x) = \frac{3}{5 + \mathrm{e}^x}$.
   Its representative curve in a coordinate system has:
   A. only one horizontal asymptote;
   B. one horizontal asymptote and one vertical asymptote;
   C. two horizontal asymptotes.
   
3. Below is the curve $\mathcal{C}_{f^{\prime\prime}}$ representing the second derivative function $f^{\prime\prime}$ of a function $f$ defined and twice differentiable on the interval $[-3.5; 6]$.
   A. The function $f$ is convex on the interval $[-3; 3]$.
   B. The function $f$ has three inflection points.
   C. The derivative function $f^{\prime}$ of $f$ is decreasing on the interval $[0; 2]$.
   
4. Consider the sequence $(u_n)$ defined for every natural integer $n$ by $u_n = n^2 - 17n + 20$.
   A. The sequence $(u_n)$ is bounded below.
   B. The sequence $(u_n)$ is decreasing.
   C. One of the terms of the sequence $(u_n)$ equals 2021.
   
5. Consider the sequence $(u_n)$ defined by $u_0 = 2$ and, for every natural integer $n$, $u_{n+1} = 0.75 u_n + 5$. Consider the following ``threshold'' function written in Python: \begin{verbatim} def seuil() : u = 2 n = 0 while u < 45 : u = 0,75*u + 5 n = n+1 return n \end{verbatim} This function returns:
   A. the smallest value of $n$ such that $u_n \geqslant 45$;
   B. the smallest value of $n$ such that $u_n < 45$;
   C. the largest value of $n$ such that $u_n \geqslant 45$.

Exercise 3
For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.

1. The sequence $(u_n)$ is defined for every natural integer $n$ by $$u_n = \frac{1 + 5^n}{2 + 3^n}$$ Statement 1: The sequence $(u_n)$ converges to $\frac{5}{3}$.
   
2. We consider the sequence $(w_n)$ defined by: $$w_0 = 0 \text{ and, for every natural integer } n,\ w_{n+1} = 3w_n - 2n + 3.$$ Statement 2: For every natural integer $n$, $w_n \geqslant n$.
   
3. We consider the function $f$ defined on $]0; +\infty[$ whose representative curve $\mathscr{C}_f$ is given in an orthonormal coordinate system in the figure (Fig. 1). We specify that:
   • $T$ is the tangent to $\mathscr{C}_f$ at point A with abscissa 8;
   • The x-axis is the horizontal tangent to $\mathscr{C}_f$ at the point with abscissa 1.
   Statement 3: According to the graph, the function $f$ is convex on its domain of definition.
   
4. Statement 4: For every real $x > 0$, $\ln(x) - x + 1 \leqslant 0$, where $\ln$ denotes the natural logarithm function.

Consider the following conditions on a function $f$ whose domain is the closed interval $[0,1]$. (For any condition involving a limit, at the endpoints, use the relevant one-sided limit.) I. $f$ is differentiable at each $x \in [0,1]$. II. $f$ is continuous at each $x \in [0,1]$. III. The set $\{f(x) \mid x \in [0,1]\}$ has a maximum element and a minimum element.
Statements
(13) If I is true, then II is true. (14) If II is true, then III is true. (15) If III is false, then I is false. (16) No two of the three given conditions are equivalent to each other. (Two statements being equivalent means each implies the other.)

Let $$f(x) = \frac{1}{|\ln x|}\left(\frac{1}{x} + \cos x\right)$$
Statements
(21) As $x \rightarrow \infty$, the sign of $f(x)$ changes infinitely many times. (22) As $x \rightarrow \infty$, the limit of $f(x)$ does not exist. (23) As $x \rightarrow 1$, $f(x) \rightarrow \infty$. (24) As $x \rightarrow 0^+$, $f(x) \rightarrow 1$.

For the function $$f ( x ) = \begin{cases} x + 2 & ( x < - 1 ) \\ 0 & ( x = - 1 ) \\ x ^ { 2 } & ( - 1 < x < 1 ) \\ x - 2 & ( x \geqq 1 ) \end{cases}$$ which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [3 points]
$\langle$Remarks$\rangle$ ㄱ. $\lim _ { x \rightarrow 1 + 0 } \{ f ( x ) + f ( - x ) \} = 0$ ㄴ. The function $f ( x ) - | f ( x ) |$ is discontinuous at 1 point. ㄷ. There is no constant $a$ such that the function $f ( x ) f ( x - a )$ is continuous on the entire set of real numbers.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the two functions $$f(x) = \begin{cases} -1 & (|x| \geq 1) \\ 1 & (|x| < 1) \end{cases}, \quad g(x) = \begin{cases} 1 & (|x| \geq 1) \\ -x & (|x| < 1) \end{cases}$$ which of the following statements are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
Remarks ᄀ. $\lim_{x \rightarrow 1} f(x)g(x) = -1$ ㄴ. The function $g(x+1)$ is continuous at $x = 0$. ㄷ. The function $f(x)g(x+1)$ is continuous at $x = -1$.
(1) ᄀ
(2) ㄴ
(3) ᄀ, ㄴ
(4) ᄀ, ㄷ
(5) ᄀ, ㄴ, ㄷ

As shown in the figure, let $\mathrm { A }$ and $\mathrm { B }$ be the $x$-intercept and $y$-intercept, respectively, of the graph of the function $y = \frac { k } { x - 1 } + 3$ where $0 < k < 3$. [Figure] Let $\mathrm { P }$ be the point (other than $\mathrm { B }$) where the line passing through the intersection of the two asymptotes of this graph and point $\mathrm { B }$ meets the graph, and let $\mathrm { Q }$ be the foot of the perpendicular from point $\mathrm { P }$ to the $x$-axis. Which of the following statements are correct? [4 points] ㄱ. When $k = 1$, the coordinates of point $\mathrm { P }$ are $( 2,4 )$. ㄴ. For real numbers $0 < k < 3$, the sum of the slope of line AB and the slope of line AP is 0. ㄷ. When the area of quadrilateral PBAQ is a natural number, the slope of line BP is between 0 and 1.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For two natural numbers $a$ and $b$, the function $f(x)$ is defined as $$f(x) = \begin{cases} 2x^3 - 6x + 1 & (x \leq 2) \\ a(x-2)(x-b) + 9 & (x > 2) \end{cases}$$ For a real number $t$, let $g(t)$ denote the number of intersection points of the graph of $y = f(x)$ and the line $y = t$. $$g(k) + \lim_{t \rightarrow k-} g(t) + \lim_{t \rightarrow k+} g(t) = 9$$ If the number of real numbers $k$ satisfying this condition is 1, find the maximum value of $a + b$ for the ordered pair $(a, b)$ of two natural numbers. [4 points]
(1) 51
(2) 52
(3) 53
(4) 54
(5) 55

15. Given functions $f ( x ) = 2 ^ { x } , g ( x ) = \hat { x } ^ { 2 } + a _ { 2 }$ (where $a \in R$). For unequal real numbers $x _ { 1 } , x _ { 2 }$, let $m = \frac { f \left( x _ { 1 } \right) - f \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } } , n = \frac { g \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$. Consider the following propositions:
(1) For any unequal real numbers $x _ { 1 } , x _ { 2 }$, we have $m > 0$; (2) For any $a$ and any unequal real numbers $x _ { 1 } , x _ { 2 }$, we have $n > 0$;
(3) For any $a$, there exist unequal real numbers $x _ { 1 } , x _ { 2 }$ such that $m = n$; (4) For any $a$, there exist unequal real numbers $x _ { 1 } , x _ { 2 }$ such that $m = - n$. The true propositions are \_\_\_\_ (write the numbers of all true propositions).
III. Solution Questions:

15. Given functions $f ( x ) = 2 ^ { x }$ and $g ( x ) = x ^ { 2 } + a x$ (where $a \in \mathbb{R}$). For unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, let $m = \frac { f \left( x _ { 1 } \right) - f \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$ and $n = \frac { g \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$.
The following propositions are given:
(1) For any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $m > 0$;
(2) For any $a$ and any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $n > 0$;
(3) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = n$;
(4) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = - n$. The true propositions are $\_\_\_\_$ (write out the numbers of all true propositions).

III. Solution Questions

Exercise III

Let $f$ be the function defined for every real number $x$ different from $1$ by $f ( x ) = \frac { 3 } { 1 - x }$ and $C _ { f }$ its representative curve in an orthonormal coordinate system. III-A- $\quad \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = - \infty$. III-B- An equation of the tangent line to the curve $C _ { f }$ at the point with abscissa $x = - 1$ is $y = \frac { 3 } { 4 } x + \frac { 3 } { 2 }$. III-C- $f$ is concave on $] 1 ; + \infty [$.
For each statement, indicate whether it is TRUE or FALSE.

Statement 1: A function $f: R \rightarrow R$ is continuous at $x_{0}$ if and only if $\lim_{x \rightarrow x_{0}} f(x)$ exists and $\lim_{x \rightarrow x_{0}} f(x) = f(x_{0})$. Statement 2: A function $f: R \rightarrow R$ is discontinuous at $x_{0}$ if and only if, $\lim_{x \rightarrow x_{0}} f(x)$ exists and $\lim_{x \rightarrow x_{0}} f(x) \neq f(x_{0})$.
(1) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.
(4) Statement 1 is true, Statement 2 is false.

Given a real-coefficient cubic polynomial function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + 3$ . Let $g ( x ) = f ( - x ) - 3$ . It is known that the graph of $y = g ( x )$ has a center of symmetry at $( 1,0 )$ and $g ( - 1 ) < 0$ . Select the correct options.
(1) $g ( x ) = 0$ has three distinct integer roots
(2) $a < 0$
(3) The center of symmetry of the graph of $y = f ( x )$ is $( - 1 , - 3 )$
(4) $f ( 100 ) < 0$
(5) The graph of $y = f ( x )$ near the point $( - 1 , f ( - 1 ) )$ can be approximated by a line with slope $a$

Let $f(x) = 2x^3 - 3x + 1$. Select the correct statements about the graph of the function $y = f(x)$.
(1) The graph of $y = f(x)$ passes through the point $(1, 0)$
(2) The graph of $y = f(x)$ has only one intersection point with the $x$-axis
(3) The point $(1, 0)$ is a center of symmetry of the graph of $y = f(x)$
(4) The graph of $y = f(x)$ approximates a straight line $y = 3x - 3$ near the center of symmetry
(5) The graph of $y = 3x^3 - 6x^2 + 2x$ can be obtained from the graph of $y = f(x)$ by appropriate translation

Let $\Gamma$ be the graph of the function $y = x ^ { 3 } - x$ on the coordinate plane. Select the correct options.
(1) The center of symmetry of $\Gamma$ is the origin
(2) $\Gamma$ approximates the line $y = x$ near $x = 0$
(3) $\Gamma$ can coincide with the graph of the function $y = x ^ { 3 } + x + 3$ after appropriate translation
(4) $\Gamma$ and the graph of the function $y = x ^ { 3 } + x$ are symmetric about the $x$-axis
(5) $\Gamma$ and the graph of the function $y = - x ^ { 3 } + x$ are symmetric about the $y$-axis

Below is the graph of the function $f$.
Accordingly, regarding the function f: I. The function f does not have an absolute maximum value on the interval $[ 0,4 ]$. II. There exists $a \in [ 0,4 ]$ such that $f ( a ) = 2$. III. $\lim _ { x \rightarrow 1 ^ { - } } ( f \circ f ) ( x ) = 1$.
Which of the following statements are true?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

A function $f$ defined on the set of real numbers satisfies the inequalities $$1 \leq f ( x ) \leq 2$$ for every $x$.
Accordingly,\ I. $\lim _ { x \rightarrow 1 } \frac { 1 } { f ( x ) }$ exists.\ II. $\lim _ { x \rightarrow 1 } \frac { f ( x ) } { x }$ exists.\ III. $\lim _ { x \rightarrow 1 } ( | f ( x ) | - f ( x ) )$ exists.
Which of the following statements are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and II\ E) II and III