Exercise 4 (7 points) Theme: numerical functions This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The six questions are independent.

1. The representative curve of the function $f$ defined on $\mathbb{R}$ by $f(x) = \frac{-2x^2 + 3x - 1}{x^2 + 1}$ admits as an asymptote the line with equation: a. $x = -2$; b. $y = -1$; c. $y = -2$; d. $y = 0$
2. Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = x\mathrm{e}^{x^2}$. The antiderivative $F$ of $f$ on $\mathbb{R}$ which satisfies $F(0) = 1$ is defined by: a. $F(x) = \frac{x^2}{2}\mathrm{e}^{x^2}$; b. $F(x) = \frac{1}{2}\mathrm{e}^{x^2}$ c. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2}$; d. $F(x) = \frac{1}{2}\mathrm{e}^{x^2} + \frac{1}{2}$
3. The representative graph $\mathscr{C}_{f'}$ of the derivative function $f'$ of a function $f$ defined on $\mathbb{R}$ is given below. We can affirm that the function $f$ is: a. concave on $]0; +\infty[$; b. convex on $]0; +\infty[$; c. convex on $[0; 2]$; d. convex on $[2; +\infty[$.
4. Among the antiderivatives of the function $f$ defined on $\mathbb{R}$ by $f(x) = 3\mathrm{e}^{-x^2} + 2$: a. all are increasing on $\mathbb{R}$; b. all are decreasing on $\mathbb{R}$; c. some are increasing on $\mathbb{R}$ and others decreasing on $\mathbb{R}$; d. all are increasing on $]-\infty; 0]$ and decreasing on $[0; +\infty[$.
5. The limit at $+\infty$ of the function $f$ defined on the interval $]0; +\infty[$ by $f(x) = \frac{2\ln x}{3x^2 + 1}$ is equal to: a. $\frac{2}{3}$; b. $+\infty$; c. $-\infty$; d. 0.
6. The equation $\mathrm{e}^{2x} + \mathrm{e}^x - 12 = 0$ admits in $\mathbb{R}$: a. three solutions; b. two solutions; c. only one solution; d. no solution.

Exercise 2 (7 points) Themes: numerical functions and sequences This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns neither points nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
For questions 1 to 3 below, consider a function $f$ defined and twice differentiable on $\mathbb{R}$. The curve of its derivative function $f'$ is given. We admit that $f'$ has a maximum at $-\frac{3}{2}$ and that its curve intersects the x-axis at the point with coordinates $\left(-\frac{1}{2}; 0\right)$.
Question 1: a. The function $f$ has a maximum at $-\frac{3}{2}$; b. The function $f$ has a maximum at $-\frac{1}{2}$; c. The function $f$ has a minimum at $-\frac{1}{2}$; d. At the point with abscissa $-1$, the curve of function $f$ has a horizontal tangent.
Question 2: a. The function $f$ is convex on $]-\infty; -\frac{3}{2}[$; c. The curve $\mathscr{C}_f$ representing function $f$ does not have an inflection point;
Question 3: The second derivative $f''$ of function $f$ satisfies: a. $f''(x) \geqslant 0$ for $x \in ]-\infty; -\frac{1}{2}[$; b. $f''(x) \geqslant 0$ for $x \in [-2; -1]$; c. $f''\left(-\frac{3}{2}\right) = 0$; d. $f''(-3) = 0$.
Question 4: Consider three sequences $\left(u_n\right)$, $\left(v_n\right)$ and $\left(w_n\right)$. We know that, for every natural number $n$, we have: $u_n \leqslant v_n \leqslant w_n$ and furthermore: $\lim_{n \rightarrow +\infty} u_n = 1$ and $\lim_{n \rightarrow +\infty} w_n = 3$. We can then affirm that: a. the sequence $\left(v_n\right)$ converges; b. If the sequence $(u_n)$ is increasing then the sequence $(v_n)$ is bounded below by $u_0$; c. $1 \leqslant v_0 \leqslant 3$; d. the sequence $(v_n)$ diverges.
Question 5: Consider a sequence $(u_n)$ such that, for every non-zero natural number $n$: $u_n \leqslant u_{n+1} \leqslant \frac{1}{n}$. We can then affirm that: a. the sequence $(u_n)$ diverges; b. the sequence $(u_n)$ converges; c. $\lim_{n \rightarrow +\infty} u_n = 0$; d. $\lim_{n \rightarrow +\infty} u_n = 1$.
Question 6: Consider $(u_n)$ a real sequence such that for every natural number $n$, we have: $n < u_n < n+1$. We can affirm that: a. There exists a natural number $N$ such that $u_N$ is an integer; b. the sequence $(u_n)$ is increasing; c. the sequence $(u_n)$ is convergent; d. The sequence $(u_n)$ has no limit.

The plane is equipped with an orthogonal coordinate system. We consider a function $f$ defined and differentiable on $\mathbb{R}$. We denote $f^{\prime}$ its derivative function. The representative curve of the derivative function $f^{\prime}$ is given.
In this part, results will be obtained by graphical reading of the representative curve of the derivative function $f^{\prime}$. No justification is required.

1. Give the direction of variation of the function $f$ on $\mathbb{R}$. Use approximate values if necessary.
2. Give the intervals on which the function $f$ appears to be convex.

The curve $\mathscr{C}$ below represents a function $f$ defined and twice differentiable on $]0; +\infty[$. We know that:

• the maximum of the function $f$ is reached at the point with abscissa 3;
• the point P with abscissa 5 is the unique inflection point of the curve $\mathscr{C}$.

We have:
A. for all $x \in ]0; 5[$, $f(x)$ and $f'(x)$ have the same sign;
C. for all $x \in ]0; 5[$, $f'(x)$ and $f''(x)$ have the same sign;
B. for all $x \in ]5; +\infty[$, $f(x)$ and $f'(x)$ have the same sign;
D. for all $x \in ]5; +\infty[$, $f(x)$ and $f''(x)$ have the same sign.

Exercise 3

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. The candidate will indicate on their answer sheet the number of the question and the chosen answer. No justification is required.
A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent. Throughout the exercise, $\mathbb { R }$ denotes the set of real numbers.

1. A primitive of the function $f$, defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$, is the function $F$, defined on $\mathbb { R }$, by: a. $F ( x ) = \frac { x ^ { 2 } } { 2 } \mathrm { e } ^ { x }$ b. $F ( x ) = ( x - 1 ) \mathrm { e } ^ { x }$ c. $F ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ d. $F ( x ) = x ^ { 2 } \mathrm { e } ^ { x ^ { 2 } }$
   
2. We consider the function $g$ defined by $g ( x ) = \ln \left( \frac { x - 1 } { 2 x + 4 } \right)$. The function $g$ is defined on: a. $\mathbb { R }$ b. $] - \infty ; - 2 [ \cup ] 1 ; + \infty [$ c. $] - 2 ; + \infty [$ d. $] - 2 ; 1 [$
   
3. The function $h$ defined on $\mathbb { R }$ by $h ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ is: a. concave on $\mathbb { R }$ b. convex on $\mathbb { R }$ c. convex on $] - \infty ; - 3 ]$ and concave on $[-3; + \infty [$ d. concave on $] - \infty ; - 3 ]$ and convex on $[-3; + \infty [$
   
4. A sequence ( $u _ { n }$ ) is bounded below by 3 and converges to a real number $\ell$. We can affirm that: a. $\ell = 3$ b. $\ell \geqslant 3$ c. The sequence ( $u _ { n }$ ) is decreasing. d. The sequence ( $u _ { n }$ ) is constant from a certain rank onwards.
   
5. The sequence ( $w _ { n }$ ) is defined by $w _ { 1 } = 2$ and for every strictly positive natural number $n$, $w _ { n + 1 } = \frac { 1 } { n } w _ { n }$. a. The sequence ( $w _ { n }$ ) is geometric b. The sequence ( $w _ { n }$ ) does not have a limit c. $w _ { 5 } = \frac { 1 } { 15 }$ d. The sequence ( $w _ { n }$ ) converges to 0.

A derivada da função $f(x) = x^3 - 3x^2 + 2x - 1$ é
(A) $f'(x) = 3x^2 - 6x + 2$ (B) $f'(x) = 3x^2 - 3x + 2$ (C) $f'(x) = x^2 - 6x + 2$ (D) $f'(x) = 3x^2 + 6x + 2$ (E) $f'(x) = 3x^2 - 6x - 2$

QUESTION 164
The derivative of $f(x) = x^3 - 4x^2 + 5x - 2$ is
(A) $f'(x) = 3x^2 - 8x + 5$
(B) $f'(x) = 3x^2 - 4x + 5$
(C) $f'(x) = 3x^2 + 8x + 5$
(D) $f'(x) = x^2 - 8x + 5$
(E) $f'(x) = 3x^2 - 8x - 5$

The derivative of $f(x) = x^3 - 3x^2 + 2x$ at $x = 1$ is:
(A) $-2$
(B) $-1$
(C) $0$
(D) $1$
(E) $2$

For the function $f ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 8$, find the value of $f ^ { \prime } ( 2 )$. [3 points]

For the function $f ( x ) = x ^ { 4 } + 3 x - 2$, what is the value of $f ^ { \prime } ( 2 )$? [3 points]
(1) 35
(2) 37
(3) 39
(4) 41
(5) 43

Let $f : (0,2) \cup (4,6) \rightarrow \mathbb{R}$ be a differentiable function. Suppose also that $f''(x) = 1$ for all $x \in (0,2) \cup (4,6)$. Which of the following is ALWAYS true?
(A) $f$ is increasing
(B) $f$ is one-to-one
(C) $f(x) = x$ for all $x \in (0,2) \cup (4,6)$
(D) $f(5.5) - f(4.5) = f(1.5) - f(0.5)$

Let $f : ( 0, 2 ) \cup ( 4, 6 ) \rightarrow \mathbb { R }$ be a differentiable function. Suppose also that $f ^ { \prime \prime } ( x ) = 1$ for all $x \in ( 0, 2 ) \cup ( 4, 6 )$. Which of the following is ALWAYS true?
(A) $f$ is increasing
(B) $f$ is one-to-one
(C) $f ( x ) = x$ for all $x \in ( 0, 2 ) \cup ( 4, 6 )$
(D) $f ( 5.5 ) - f ( 4.5 ) = f ( 1.5 ) - f ( 0.5 )$

Consider the function $f : \mathbb{R} \longrightarrow \mathbb{R}$, defined as follows: $$f(x) = \begin{cases} (x-1)\min\left\{x, x^2\right\} & \text{if } x \geq 0 \\ x\min\left\{x, \frac{1}{x}\right\} & \text{if } x < 0 \end{cases}$$ Then, $f$ is
(A) differentiable everywhere.
(B) not differentiable at exactly one point.
(C) not differentiable at exactly two points.
(D) not differentiable at exactly three points.

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $( 2 + a ) ^ { 2 } f ^ { \prime \prime } ( 1 ) + ( 2 - a ) ^ { 2 } f ^ { \prime \prime } ( - 1 ) = 0$
(B) $( 2 - a ) ^ { 2 } f ^ { \prime \prime } ( 1 ) - ( 2 + a ) ^ { 2 } f ^ { \prime \prime } ( - 1 ) = 0$
(C) $f ^ { \prime } ( 1 ) f ^ { \prime } ( - 1 ) = ( 2 - a ) ^ { 2 }$
(D) $f ^ { \prime } ( 1 ) f ^ { \prime } ( - 1 ) = - ( 2 + a ) ^ { 2 }$

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 $f ^ { \prime \prime } ( x ) - 2 f ^ { \prime } ( x ) + f ( x ) \geq e ^ { x } , x \in [ 0,1 ]$.
If the function $\mathrm { e } ^ { - x } f ( x )$ assumes its minimum in the interval $[ 0,1 ]$ at $x = \frac { 1 } { 4 }$, which of the following is true?
(A) $f ^ { \prime } ( x ) < f ( x ) , \frac { 1 } { 4 } < x < \frac { 3 } { 4 }$
(B) $f ^ { \prime } ( x ) > f ( x ) , \quad 0 < x < \frac { 1 } { 4 }$
(C) $f ^ { \prime } ( x ) < f ( x ) , \quad 0 < x < \frac { 1 } { 4 }$
(D) $f ^ { \prime } ( x ) < f ( x ) , \frac { 3 } { 4 } < x < 1$

Let $f ( x ) = \min \{ 1,1 + x \sin x \} , 0 \leq x \leq 2 \pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not continuous, then the ordered pair $( m , n )$ is equal to
(1) $( 2,0 )$
(2) $( 1,0 )$
(3) $( 1,1 )$
(4) $( 2,1 )$

The number of points, where the function $f : R \rightarrow R , f ( x ) = | x - 1 | \cos | x - 2 | \sin | x - 1 | + ( x - 3 ) \left| x ^ { 2 } - 5 x + 4 \right|$, is NOT differentiable, is
(1) 1
(2) 2
(3) 3
(4) 4

On the coordinate plane, consider the graphs of two functions $f(x) = x^{5} - 5x^{3} + 5x^{2} + 5$ and $g(x) = \sin\left(\frac{\pi x}{3} + \frac{\pi}{2}\right)$ (where $\pi$ is the circumference ratio). Select the correct options.
(1) $f'(1) = 0$
(2) $y = f(x)$ is increasing on the closed interval $[0, 2]$
(3) $y = f(x)$ is concave up on the closed interval $[0, 2]$
(4) For any real number $x$, $g(x + 6\pi) = g(x)$
(5) Both $y = f(x)$ and $y = g(x)$ are increasing on the closed interval $[3, 4]$

Below is the graph of the derivative of a function f defined on the interval $[ - 5,5 ]$.
According to this graph, I. The function f is decreasing for $x > 0$. II. $f ( - 2 ) > f ( 0 ) > f ( 2 )$. III. The function f has local extrema at $x = - 2$ and $x = 2$. Which of the following statements are true?
A) Only I
B) Only II
C) I and II
D) I and III
E) I, II and III

Below, the graph of the derivative of a function f that is defined and continuous on the set of real numbers is given.
Accordingly,
I. $f ( 2 ) - f ( 1 ) = -2$. II. The function f has a local maximum at the point $x = 0$. III. The second derivative function is defined at the point $x = 0$.
Which of the following statements are true?
A) Only I
B) Only III
C) I and II
D) II and III
E) I, II and III

$$f ( x ) = 2 x ( x - 1 ) ^ { 3 } + ( x - 1 ) ^ { 4 }$$
What is the value of the third derivative of the function at the point $x = 1$?
A) 10
B) 12
C) 14
D) 16
E) 18

A function f is continuous on the closed interval $[ 0,6 ]$ and differentiable on each of the open intervals $( 0,3 ) , ( 3,4 ) , ( 4,6 )$. The graph of its derivative $f ^ { \prime }$ is given in the rectangular coordinate plane below.
$$\begin{gathered} \text{Let } 0 < c < 2 \text{ and } \\ f ( 0 ) = 5 \end{gathered}$$
Accordingly, which of the following could be the value of f(6)?
A) 5,5
B) 7,3
C) 10,1
D) 12,7
E) 14,9