$\lim _ { x \rightarrow \infty } \frac { \sqrt { 9 x ^ { 4 } + 1 } } { x ^ { 2 } - 3 x + 5 }$ is
(A) 1
(B) 3
(C) 9
(D) nonexistent

The graph of the piecewise-defined function $f$ is shown in the figure above. The graph has a vertical tangent line at $x = - 2$ and horizontal tangent lines at $x = - 3$ and $x = - 1$. What are all values of $x , - 4 < x < 3$, at which $f$ is continuous but not differentiable?
(A) $x = 1$
(B) $x = - 2$ and $x = 0$
(C) $x = - 2$ and $x = 1$
(D) $x = 0$ and $x = 1$

The graph of the function $f$ is shown in the figure above. The value of $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ is
(A) - 2
(B) - 1
(C) 2
(D) nonexistent

Let $f$ be the function defined by $f ( x ) = \sqrt { | x - 2 | }$ for all $x$. Which of the following statements is true?
(A) $f$ is continuous but not differentiable at $x = 2$.
(B) $f$ is differentiable at $x = 2$.
(C) $f$ is not continuous at $x = 2$.
(D) $\lim _ { x \rightarrow 2 } f ( x ) \neq 0$
(E) $x = 2$ is a vertical asymptote of the graph of $f$.

The figure above shows the graph of $f$. If $f ( x ) = \int _ { 2 } ^ { x } g ( t ) d t$, which of the following could be the graph of $y = g ( x )$ ?
(A) [graph A]
(B) [graph B]
(C) [graph C]
(D) [graph D]
(E) [graph E]

The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

Let $f$ be a function that is continuous on the closed interval $[ 2,4 ]$ with $f ( 2 ) = 10$ and $f ( 4 ) = 20$. Which of the following is guaranteed by the Intermediate Value Theorem?
(A) $f ( x ) = 13$ has at least one solution in the open interval $( 2,4 )$.
(B) $f ( 3 ) = 15$
(C) $f$ attains a maximum on the open interval $( 2,4 )$.
(D) $f ^ { \prime } ( x ) = 5$ has at least one solution in the open interval $( 2,4 )$.
(E) $f ^ { \prime } ( x ) > 0$ for all $x$ in the open interval $( 2,4 )$.

The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

In the plane equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, we denote by $\mathscr{C}_{u}$ the curve representing the function $u$ defined on the interval $]0; +\infty[$ by: $$u(x) = a + \frac{b}{x} + \frac{c}{x^2}$$ where $a, b$ and $c$ are fixed real numbers.
The curve $\mathscr{C}_{u}$ passes through the points $\mathrm{A}(1; 0)$ and $\mathrm{B}(4; 0)$ and the $y$-axis and the line $\mathscr{D}$ with equation $y = 1$ are asymptotes to the curve $\mathscr{C}_{u}$.

1. Give the values of $u(1)$ and $u(4)$.
2. Give $\lim_{x \rightarrow +\infty} u(x)$. Deduce the value of $a$.
3. Deduce that, for all strictly positive real $x$, $u(x) = \frac{x^2 - 5x + 4}{x^2}$.

Throughout the exercise, $n$ denotes a strictly positive natural number. The purpose of the exercise is to study the equation
$$\left( E _ { n } \right) : \quad \frac { \ln ( x ) } { x } = \frac { 1 } { n }$$
with unknown strictly positive real number $x$.

Part A

Let $f$ be the function defined on the interval $] 0$; $+ \infty [$ by
$$f ( x ) = \frac { \ln ( x ) } { x }$$
It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.

1. Study the variations of function $f$.
2. Determine its maximum.

Part B

1. Show that, for $n \geqslant 3$, the equation $f ( x ) = \frac { 1 } { n }$ has a unique solution on $[ 1 ; e]$ denoted $\alpha _ { n }$.
2. From the above, for every integer $n \geqslant 3$, the real number $\alpha _ { n }$ is a solution of equation $\left( E _ { n } \right)$. a. On the graph are drawn the lines $D _ { 3 } , D _ { 4 }$ and $D _ { 5 }$ with equations respectively $y = \frac { 1 } { 3 } , y = \frac { 1 } { 4 }$, $y = \frac { 1 } { 5 }$. Conjecture the direction of variation of the sequence ( $\alpha _ { n }$ ). b. Compare, for every integer $n \geqslant 3 , f \left( \alpha _ { n } \right)$ and $f \left( \alpha _ { n + 1 } \right)$. Determine the direction of variation of the sequence $\left( \alpha _ { n } \right)$. c. Deduce that the sequence ( $\alpha _ { n }$ ) converges. It is not asked to calculate its limit.
3. It is admitted that, for every integer $n \geqslant 3$, equation $\left( E _ { n } \right)$ has another solution $\beta _ { n }$ such that $$1 \leqslant \alpha _ { n } \leqslant \mathrm { e } \leqslant \beta _ { n }$$ a. It is admitted that the sequence ( $\beta _ { n }$ ) is increasing. Establish that, for every natural number $n$ greater than or equal to 3, $$\beta _ { n } \geqslant n \frac { \beta _ { 3 } } { 3 } .$$ b. Deduce the limit of the sequence ( $\beta _ { n }$ ).

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 — 5 points Theme: function study Parts A and B can be treated independently
Part A
The plane is equipped with an orthogonal coordinate system. Below is represented the curve of a function $f$ defined and twice differentiable on $\mathbb{R}$, as well as that of its derivative $f'$ and its second derivative $f''$.

1. Determine, by justifying your choice, which curve corresponds to which function.
2. Determine, with the precision allowed by the graph, the slope of the tangent line to curve $\mathscr{C}_{2}$ at the point with abscissa 4.
3. Give, with the precision allowed by the graph, the abscissa of each inflection point of curve $\mathscr{C}_{1}$.

Part B
Let $k$ be a strictly positive real number. We consider the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \frac{4}{1 + \mathrm{e}^{-kx}}$$

1. Determine the limits of $g$ at $+\infty$ and at $-\infty$.
2. Prove that $g'(0) = k$.
3. By admitting the result below obtained with computer algebra software, prove that the curve of $g$ has an inflection point at the point with abscissa 0.

$\triangleright$	Computer algebra
	$g(x) = 4 / (1 + \mathrm{e}^{\wedge}(-kx))$
1
	$\rightarrow g(x) = \frac{4}{\mathrm{e}^{-kx} + 1}$
	Simplify $(g''(x))$
2
	$\rightarrow g''(x) = -4\mathrm{e}^{kx}(\mathrm{e}^{kx} - 1)\frac{k^{2}}{(\mathrm{e}^{kx} + 1)^{3}}$

We consider a function $f$ defined on $[0; +\infty[$, represented by the curve $\mathscr{C}$ below. The line $T$ is tangent to the curve $\mathscr{C}$ at point A with abscissa $\frac{5}{2}$.

1. Draw up, by graphical reading, the table of variations of the function $f$ on the interval $[0;5]$.
2. What does the curve $\mathscr{C}$ appear to present at point A?
3. The derivative $f'$ and the second derivative $f''$ of the function $f$ are represented by the curves $\mathscr{C}_1$ and $\mathscr{C}_2$. Associate with each of these two functions the curve that represents it. This choice will be justified.
4. Can the curve $\mathscr{C}_3$ be the graphical representation on $[0; +\infty[$ of a primitive of the function $f$? Justify.

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.

Question 144
A figura mostra o gráfico de uma função $f$.
[Figure]
Com base no gráfico, é correto afirmar que
(A) $f(-2) = 0$ (B) $f(0) = -2$ (C) $f(1) = 3$ (D) $f(2) = 0$ (E) $f(3) = 1$

During a Mathematics class, the teacher suggests to the students that a Cartesian coordinate system $(x, y)$ be established and represents on the board the description of five algebraic sets, I, II, III, IV and V, as follows:
I - is the circle with equation $x^{2} + y^{2} = 9$; II - is the parabola with equation $y = -x^{2} - 1$, with $x$ varying from $-1$ to $1$; III - is the square formed by the vertices $(-2,1)$, $(-1,1)$, $(-1,2)$ and $(-2,2)$; IV - is the square formed by the vertices $(1,1)$, $(2,1)$, $(2,2)$ and $(1,2)$; V - is the point $(0,0)$.
Next, the teacher correctly represents the five sets on the same grid, composed of squares with sides measuring one unit of length each, obtaining a figure.
Which of these figures was drawn by the teacher?
(A), (B), (C), (D), (E) [as shown in the figures]

Some electronic equipment can ``burn out'' during operation when its internal temperature reaches a maximum value $\mathrm{T}_{\mathrm{M}}$. For greater durability of its products, the electronics industry connects temperature sensors to this equipment, which activate an internal cooling system, turning it on when the temperature of the electronic device exceeds a critical level $\mathrm{T}_{\mathrm{c}}$, and turning it off only when the temperature drops to values below $\mathrm{T}_{\mathrm{m}}$. The graph illustrates the oscillation of the internal temperature of an electronic device during the first six hours of operation, showing that its internal cooling system was activated several times.
How many times did the temperature sensor activate the system, turning it on or off?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 9

A pharmaceutical company conducted a study of the efficacy (in percentage) of a medication over 12 hours of treatment in a patient. The medication was administered in two doses, with a 6-hour interval between them. As soon as the first dose was administered, the medication's efficacy increased linearly for 1 hour, until reaching maximum efficacy (100\%), and remained at maximum efficacy for 2 hours. After these 2 hours at maximum efficacy, it began to decrease linearly, reaching 20\% efficacy upon completing the initial 6 hours of analysis. At this moment, the second dose was administered, which began to increase linearly, reaching maximum efficacy after 0.5 hours and remaining at 100\% for 3.5 hours. In the remaining hours of analysis, the efficacy decreased linearly, reaching 50\% efficacy at the end of treatment.
Considering the quantities time (in hours) on the horizontal axis and medication efficacy (in percentage) on the vertical axis, which graph represents this study?
(A), (B), (C), (D), (E) [see figures]

The English physiologist Archibald Vivian Hill proposed, in his studies, that the velocity $v$ of contraction of a muscle when subjected to a weight $p$ is given by the equation $( p + a )( v + b ) = K$, with $a$, $b$, and $K$ constants.
A physiotherapist, with the intention of maximizing the beneficial effect of the exercises he would recommend to one of his patients, wanted to study this equation and classified it as follows:

Type of curve

Oblique half-line

Horizontal half-line

Branch of parabola

Arc of circle

Branch of hyperbola

The physiotherapist analyzed the dependence between $v$ and $p$ in Hill's equation and classified it according to its geometric representation in the Cartesian plane, using the coordinate pair ($p$; $v$). Assume that $K > 0$.
The graph of the equation that the physiotherapist used to maximize the effect of the exercises is of the type
(A) oblique half-line.
(B) horizontal half-line.
(C) branch of parabola.
(D) arc of circle.
(E) branch of hyperbola.

The Church of Saint Francis of Assisi, a modernist architectural work by Oscar Niemeyer, located at Pampulha Lake, in Belo Horizonte, has parabolic vaults. Figure 2 provides a front view of one of the vaults, with hypothetical measurements to simplify the calculations.
What is the measure of the height H, in meters, indicated in Figure 2?
(A) $\frac{16}{3}$
(B) $\frac{31}{5}$
(C) $\frac{25}{4}$
(D) $\frac{25}{3}$
(E) $\frac{75}{2}$

Research in the area of neurobiology confirms that meditative practice is responsible for considerably reducing respiratory frequency for advanced practitioners, who, after initiating meditation, have their respiratory frequencies reduced until they stabilize at a lower level. The graph presents the relationship of respiratory frequency, in breaths per minute (rpm), in relation to time, in minutes, of an advanced practitioner, in which $(\mathrm{f}_1)$ represents the frequency at instant $\mathrm{t}_1$, when meditative practice begins; and $(\mathrm{f}_2)$, the frequency at instant $t_2$, from which it stabilizes during meditation.
From the instant $\mathrm{t}_1$, when the meditative practice begins, the behavior of respiratory frequency, in relation to time,
(A) remains constant.
(B) is directly proportional to time.
(C) is inversely proportional to time.
(D) decreases until the instant $\mathrm{t}_2$, after which it becomes constant.
(E) decreases proportionally to time, both between $\mathrm{t}_1$ and $\mathrm{t}_2$ and after $t_2$.

Let $$f ( x ) = \frac { x ^ { 4 } } { ( x - 1 ) ( x - 2 ) \cdots ( x - n ) }$$ where the denominator is a product of $n$ factors, $n$ being a positive integer. It is also given that the $X$-axis is a horizontal asymptote for the graph of $f$. Answer the independent questions below by choosing the correct option from the given ones. a) How many vertical asymptotes does the graph of $f$ have?
Options: $n$ less than $n$ more than $n$ impossible to decide
Answer: $\_\_\_\_$ b) What can you deduce about the value of $n$?
Options: $n < 4$ $n = 4$ $n > 4$ impossible to decide
Answer: $\_\_\_\_$ c) As one travels along the graph of $f$ from left to right, at which of the following points is the sign of $f ( x )$ guaranteed to change from positive to negative?
Options: $x = 0$ $x = 1$ $x = n - 1$ $x = n$
Answer: $\_\_\_\_$ d) How many inflection points does the graph of $f$ have in the region $x < 0$?
Options: none 1 more than 1 impossible to decide
(Hint: Sketching is better than calculating.)
Answer: $\_\_\_\_$

Write your answers to each question below as a series of three letters Y (for Yes) or N (for No). Leave space between the group of three letters answering (i), the answers to (ii) and the answers to (iii). Consider the graphs of functions $$f(x) = \frac{x^{3}}{x^{2}-x} \qquad g(x) = \frac{x^{2}-x}{x^{3}} \qquad h(x) = \frac{x^{3}-x}{x^{3}+x}$$ (i) Does $f$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(ii) Does $g$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(iii) Does $h$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?

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 $x \rightarrow 0^+$, $f(x) \rightarrow 1$.