The velocity of a particle moving along a straight line is given by $v ( t ) = 1.3 t \ln ( 0.2 t + 0.4 )$ for time $t \geq 0$. What is the acceleration of the particle at time $t = 1.2$?
(A) - 0.580
(B) - 0.548
(C) - 0.093
(D) 0.660

Let $f$ be a function defined by $$f(x) = \begin{cases} 1 - 2\sin x & \text{for } x \leq 0 \\ e^{-4x} & \text{for } x > 0. \end{cases}$$
(a) Show that $f$ is continuous at $x = 0$.
(b) For $x \neq 0$, express $f'(x)$ as a piecewise-defined function. Find the value of $x$ for which $f'(x) = -3$.
(c) Find the average value of $f$ on the interval $[-1, 1]$.

If $f ( x ) = 7 x - 3 + \ln x$, then $f ^ { \prime } ( 1 ) =$
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Consider the function $f$ defined on $[0; +\infty[$ by $$f(x) = \frac{\ln(x + 3)}{x + 3}$$

1. Show that $f$ is differentiable on $[0; +\infty[$. Study the sign of its derivative function $f'$, its possible limit at $+\infty$, and draw up the table of its variations.
2. We define the sequence $(u_n)_{n \geqslant 0}$ by its general term $u_n = \int_n^{n+1} f(x)\,\mathrm{d}x$. a. Justify that, if $n \leqslant x \leqslant n+1$, then $f(n+1) \leqslant f(x) \leqslant f(n)$. b. Show, without attempting to calculate $u_n$, that, for every natural integer $n$, $$f(n+1) \leqslant u_n \leqslant f(n).$$ c. Deduce that the sequence $(u_n)$ is convergent and determine its limit.
3. Let $F$ be the function defined on $[0; +\infty[$ by $$F(x) = [\ln(x+3)]^2.$$ a. Justify the differentiability on $[0; +\infty[$ of the function $F$ and determine, for every positive real $x$, the number $F'(x)$. b. We set, for every natural integer $n$, $I_n = \int_0^n f(x)\,\mathrm{d}x$. Calculate $I_n$.
4. We set, for every natural integer $n$, $S_n = u_0 + u_1 + \cdots + u_{n-1}$. Calculate $S_n$. Is the sequence $(S_n)$ convergent?

Exercise 2 -- Common to all candidates
Consider the functions $f$ and $g$ defined for all real $x$ by: $$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 1 - \mathrm{e}^{-x}.$$ The representative curves of these functions in an orthogonal coordinate system of the plane, denoted respectively $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$, are provided in the appendix.
Part A
These curves appear to admit two common tangent lines. Draw these tangent lines as accurately as possible on the figure in the appendix.
Part B
In this part, the existence of these common tangent lines is admitted. Let $\mathscr{D}$ denote one of them. This line is tangent to the curve $\mathscr{C}_{f}$ at point A with abscissa $a$ and tangent to the curve $\mathscr{C}_{g}$ at point B with abscissa $b$.

1. a. Express in terms of $a$ the slope of the tangent line to the curve $\mathscr{C}_{f}$ at point A. b. Express in terms of $b$ the slope of the tangent line to the curve $\mathscr{C}_{g}$ at point B. c. Deduce that $b = -a$.
2. Prove that the real number $a$ is a solution of the equation $$2(x - 1)\mathrm{e}^{x} + 1 = 0.$$

Part C
Consider the function $\varphi$ defined on $\mathbb{R}$ by $$\varphi(x) = 2(x - 1)\mathrm{e}^{x} + 1$$

1. a. Calculate the limits of the function $\varphi$ at $-\infty$ and $+\infty$. b. Calculate the derivative of the function $\varphi$, then study its sign. c. Draw the variation table of the function $\varphi$ on $\mathbb{R}$. Specify the value of $\varphi(0)$.
2. a. Prove that the equation $\varphi(x) = 0$ has exactly two solutions in $\mathbb{R}$. b. Let $\alpha$ denote the negative solution of the equation $\varphi(x) = 0$ and $\beta$ the positive solution of this equation. Using a calculator, give the values of $\alpha$ and $\beta$ rounded to the nearest hundredth.

Part D
In this part, we prove the existence of these common tangent lines, which was admitted in Part B. Let E be the point on the curve $\mathscr{C}_{f}$ with abscissa $\alpha$ and F the point on the curve $\mathscr{C}_{g}$ with abscissa $-\alpha$ ($\alpha$ is the real number defined in Part C).

1. Prove that the line $(EF)$ is tangent to the curve $\mathscr{C}_{f}$ at point E.
2. Prove that $(EF)$ is tangent to $\mathscr{C}_{g}$ at point F.

Let $f$ be the function defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \frac { 1 + \ln ( x ) } { x ^ { 2 } }$$
and let $\mathscr { C }$ be the representative curve of the function $f$ in a coordinate system of the plane.

1. a. Study the limit of $f$ at 0. b. What is $\lim _ { x \rightarrow + \infty } \frac { \ln ( x ) } { x }$ ? Deduce the limit of the function $f$ at $+ \infty$. c. Deduce the possible asymptotes to the curve $\mathscr { C }$.
2. a. Let $f ^ { \prime }$ denote the derivative function of the function $f$ on the interval $] 0 ; + \infty [$. Prove that, for every real $x$ belonging to the interval $] 0 ; + \infty [$, $$f ^ { \prime } ( x ) = \frac { - 1 - 2 \ln ( x ) } { x ^ { 3 } }$$ b. Solve on the interval $] 0 ; + \infty [$ the inequality $- 1 - 2 \ln ( x ) > 0$. Deduce the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$. c. Draw up the table of variations of the function $f$.
3. a. Prove that the curve $\mathscr { C }$ has a unique point of intersection with the $x$-axis, whose coordinates you will specify. b. Deduce the sign of $f ( x )$ on the interval $] 0 ; + \infty [$.
4. For every integer $n \geqslant 1$, we denote by $I _ { n }$ the area, expressed in square units, of the region bounded by the $x$-axis, the curve $\mathscr { C }$ and the lines $x = 1$ and $x = n$.

On the graph below, we have drawn, in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, a curve $\mathscr{C}$ and the line $(\mathrm{AB})$ where A and B are the points with coordinates $(0;1)$ and $(-1;3)$ respectively.
We denote by $f$ the function differentiable on $\mathbb{R}$ whose representative curve is $\mathscr{C}$. We further assume that there exists a real number $a$ such that for all real $x$, $$f(x) = x + 1 + ax\mathrm{e}^{-x^{2}}$$

1. a. Justify that the curve $\mathscr{C}$ passes through point A. b. Determine the slope of the line (AB). c. Prove that for all real $x$, $$f^{\prime}(x) = 1 - a\left(2x^{2} - 1\right)\mathrm{e}^{-x^{2}}$$ d. We assume that the line $(\mathrm{AB})$ is tangent to the curve $\mathscr{C}$ at point A. Determine the value of the real number $a$.
2. According to the previous question, for all real $x$, $$f(x) = x + 1 - 3x\mathrm{e}^{-x^{2}} \text{ and } f^{\prime}(x) = 1 + 3\left(2x^{2} - 1\right)\mathrm{e}^{-x^{2}}.$$ a. Prove that for all real $x$ in the interval $]-1;0]$, $f(x) > 0$. b. Prove that for all real $x$ less than or equal to $-1$, $f^{\prime}(x) > 0$. c. Prove that there exists a unique real number $c$ in the interval $\left[-\frac{3}{2};-1\right]$ such that $f(c) = 0$. Justify that $c < -\frac{3}{2} + 2 \cdot 10^{-2}$.
3. We denote by $\mathscr{A}$ the area, expressed in square units, of the region defined by: $$c \leqslant x \leqslant 0 \quad \text{and} \quad 0 \leqslant y \leqslant f(x)$$ a. Write $\mathscr{A}$ in the form of an integral. b. We admit that the integral $I = \int_{-\frac{3}{2}}^{0} f(x)\,\mathrm{d}x$ is an approximate value of $\mathscr{A}$ to within $10^{-3}$. Calculate the exact value of the integral $I$.

For every natural number $n$, the function $f_{n}$ is defined for all real $x$ in the interval $[0; 1]$ by:
$$f_{n}(x) = x + \mathrm{e}^{n(x-1)}$$
Let $\mathscr{C}_{n}$ denote the graph of the function $f_{n}$ in an orthogonal coordinate system.

Part A: generalities on the functions $\boldsymbol{f}_{\boldsymbol{n}}$

1. Prove that, for every natural number $n$, the function $f_{n}$ is increasing and positive on the interval $[0; 1]$.
2. Show that the curves $\mathscr{C}_{n}$ all have a common point A, and specify its coordinates.
3. Using the graphical representations, can one conjecture the behavior of the slopes of the tangent lines at A to the curves $\mathscr{C}_{n}$ for large values of $n$? Prove this conjecture.

Part B: evolution of $\boldsymbol{f}_{\boldsymbol{n}}(\boldsymbol{x})$ when $x$ is fixed

Let $x$ be a fixed real number in the interval $[0; 1]$. For every natural number $n$, we set $u_{n} = f_{n}(x)$.

1. In this question, assume that $x = 1$. Study the possible limit of the sequence $(u_{n})$.
2. In this question, assume that $0 \leq x < 1$. Study the possible limit of the sequence $(u_{n})$.

Part C: area under the curves $\mathscr{C}_{\boldsymbol{n}}$

For every natural number $n$, let $A_{n}$ denote the area, expressed in square units, of the region located between the $x$-axis, the curve $\mathscr{C}_{n}$ and the lines with equations $x = 0$ and $x = 1$ respectively. Based on the graphical representations, conjecture the limit of the sequence $(A_{n})$ as the integer $n$ tends to $+\infty$, then prove this conjecture.

The profile of a slide is modelled by the curve $\mathcal { C }$ representing the function $f$ defined on the interval [1;8] by
$$f ( x ) = ( a x + b ) \mathrm { e } ^ { - x } \text { where } a \text { and } b \text { are two natural integers. }$$
The curve $\mathcal { C }$ is drawn in an orthonormal coordinate system with unit of one metre.

Part A Modelling

1. We want the tangent to the curve $\mathcal { C }$ at its point with abscissa 1 to be horizontal. Determine the value of the integer $b$.
2. We want the top of the slide to be located between 3.5 and 4 metres high. Determine the value of the integer $a$.

Part B An amenity for visitors

We assume in the following that the function $f$ introduced in Part A is defined for all real $x \in [ 1 ; 8 ]$ by
$$f ( x ) = 10 x \mathrm { e } ^ { - x }$$
The retaining wall of the slide will be painted by an artist on a single face. In the quote he proposes, he asks for a flat fee of 300 euros plus 50 euros per square metre painted.

1. Let $g$ be the function defined on [ $1 ; 8$ ] by
   $$g ( x ) = 10 ( - x - 1 ) \mathrm { e } ^ { - x }$$
   Determine the derivative of the function $g$.
2. What is the amount of the artist's quote?

Part C A constraint to verify

Safety considerations require limiting the maximum slope of the slide. Consider a point $M$ on the curve $\mathcal { C }$, with abscissa different from 1. We call $\alpha$ the acute angle formed by the tangent to $\mathcal { C }$ at $M$ and the horizontal axis. The constraints require that the angle $\alpha$ be less than 55 degrees.

1. We denote $f ^ { \prime }$ the derivative of the function $f$ on the interval $[ 1 ; 8 ]$. We admit that, for all $x$ in the interval $[ 1 ; 8 ] , f ^ { \prime } ( x ) = 10 ( 1 - x ) \mathrm { e } ^ { - x }$. Study the variations of the function $f ^ { \prime }$ on the interval [ $1 ; 8$ ].
2. Let $x$ be a real number in the interval ] 1; 8] and let $M$ be the point with abscissa $x$ on the curve $\mathcal { C }$. Justify that $\tan \alpha = \left| f ^ { \prime } ( x ) \right|$.
3. Is the slide compliant with the imposed constraints?

The curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ given in appendix 1 are the graphical representations, in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ), of two functions $f$ and $g$ defined on $[ 0 ; + \infty [$. We consider the points $\mathrm { A } ( 0,5 ; 1 )$ and $\mathrm { B } ( 0 ; - 1 )$ in the coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ). We know that O belongs to $\mathscr { C } _ { f }$ and that the line (OA) is tangent to $\mathscr { C } _ { f }$ at point O.

1. We assume that the function $f$ is written in the form $f ( x ) = ( a x + b ) \mathrm { e } ^ { - x ^ { 2 } }$ where $a$ and $b$ are real numbers. Determine the exact values of the real numbers $a$ and $b$, detailing the approach. From now on, we consider that $\boldsymbol { f } ( \boldsymbol { x } ) = \mathbf { 2 } \boldsymbol { x } \mathrm { e } ^ { - \boldsymbol { x } ^ { \mathbf { 2 } } }$ for all $\boldsymbol { x }$ belonging to $[ \mathbf { 0 } ; + \infty [$
2. a. We will admit that, for all real $x$ strictly positive, $f ( x ) = \frac { 2 } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$.
   Calculate $\lim _ { x \rightarrow + \infty } f ( x )$. b. Draw up, justifying it, the table of variations of the function $f$ on $[ 0 ; + \infty [$.
3. The function $g$ whose representative curve $\mathscr { C } _ { g }$ passes through the point $\mathrm { B } ( 0 ; - 1 )$ is a primitive of the function $f$ on $[ 0 ; + \infty [$. a. Determine the expression of $g ( x )$. b. Let $m$ be a strictly positive real number.
   Calculate $I _ { m } = \int _ { 0 } ^ { m } f ( t ) \mathrm { d } t$ as a function of $m$. c. Determine $\lim _ { m \rightarrow + \infty } I _ { m }$.
4. a. Justify that $f$ is a probability density function on $[ 0 ; + \infty [$. b. Let $X$ be a continuous random variable that admits the function $f$ as its probability density function. Justify that, for all real $x$ in $[ 0 ; + \infty [$, $P ( X \leqslant x ) = g ( x ) + 1$. c. Deduce the exact value of the real number $\alpha$ such that $P ( X \leqslant \alpha ) = 0,5$. d. Without using an approximate value of $\alpha$, construct in the coordinate system of appendix 1 the point with coordinates ( $\alpha ; 0$ ) leaving the construction lines visible. Then shade the region of the plane corresponding to $P ( X \leqslant \alpha )$.

Exercise 1 - Part A

Here are two curves $\mathcal { C } _ { 1 }$ and $\mathcal { C } _ { 2 }$ which give for two people $P _ { 1 }$ and $P _ { 2 }$ of different body compositions the concentration $C$ of alcohol in the blood (blood alcohol level) as a function of time $t$ after ingestion of the same quantity of alcohol. The instant $t = 0$ corresponds to the moment when the two individuals ingest the alcohol. $C$ is expressed in grams per litre and $t$ in hours.

1. The function $C$ is defined on the interval $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $C ^ { \prime }$ its derivative function. At an instant $t$ positive or zero, the rate of appearance of alcohol in the blood is given by $C ^ { \prime } ( t )$. At what instant is this rate maximal? It is often said that a person of weak body composition experiences the effects of alcohol more quickly.
2. On the previous graph, identify the curve corresponding to the person with the largest body composition. Justify the choice made.
3. A person on an empty stomach ingests alcohol. It is admitted that the concentration $C$ of alcohol in their blood can be modelled by the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( t ) = A t \mathrm { e } ^ { - t }$$ where $A$ is a positive constant that depends on the body composition and the quantity of alcohol ingested. a. We denote $f ^ { \prime }$ the derivative function of the function $f$. Determine $f ^ { \prime } ( 0 )$. b. Is the following statement true? ``For equal quantities of alcohol ingested, the larger $A$ is, the more corpulent the person is.''

Part B - A particular case

Paul, a 19-year-old student of average body composition and a young driver, drinks two glasses of rum. The concentration $C$ of alcohol in his blood is modelled as a function of time $t$, expressed in hours, by the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( t ) = 2 t \mathrm { e } ^ { - t } .$$

1. Study the variations of the function $f$ on the interval $[ 0 ; + \infty [$.
2. At what instant is the concentration of alcohol in Paul's blood maximal? What is its value then? Round to $10 ^ { - 2 }$ near.
3. Recall the limit of $\frac { \mathrm { e } ^ { t } } { t }$ as $t$ tends to $+ \infty$ and deduce from it that of $f ( t )$ at $+ \infty$. Interpret the result in the context of the exercise.
4. Paul wants to know after how much time he can take his car. We recall that the legislation allows a maximum concentration of alcohol in the blood of $0,2 \mathrm {~g} . \mathrm { L } ^ { - 1 }$ for a young driver. a. Prove that there exist two real numbers $t _ { 1 }$ and $t _ { 2 }$ such that $f \left( t _ { 1 } \right) = f \left( t _ { 2 } \right) = 0,2$. b. What minimum duration must Paul wait before he can take the wheel in full compliance with the law? Give the result rounded to the nearest minute.
5. The minimum concentration of alcohol detectable in the blood is estimated at $5 \times 10 ^ { - 3 }$ g.L${}^{ - 1 }$. a. Justify that there exists an instant $T$ from which the concentration of alcohol in the blood is no longer detectable. b. The following algorithm is given where $f$ is the function defined by $f ( t ) = 2 t \mathrm { e } ^ { - t }$.
   

   Initialization:	$t$ takes the value 3,5
   	$p$ takes the value 0,25
   	$C$ takes the value 0,21
   Processing:	While $C > 5 \times 10 ^ { - 3 }$ do:
   	$\quad \mid \quad t$ takes the value $t + p$
   	$\quad C$ takes the value $f ( t )$
   Output:	End While
   Display $t$

   
   Copy and complete the following table of values by executing this algorithm. Round the values to $10 ^ { - 2 }$ near.
   

   	Initialization	Step 1	Step 2
   $p$	0,25
   $t$	3,5
   $C$	0,21

   
   What does the value displayed by this algorithm represent?

We consider the function $f$ defined for all real $x$ by $f(x) = x\mathrm{e}^{1-x^{2}}$.

1. Calculate the limit of the function $f$ at $+\infty$. Hint: you may use the fact that for all real $x$ different from 0, $f(x) = \frac{\mathrm{e}}{x} \times \frac{x^{2}}{\mathrm{e}^{x^{2}}}$. It is admitted that the limit of the function $f$ at $-\infty$ is equal to 0.
2. a. It is admitted that $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative. Prove that for all real $x$, $$f'(x) = \left(1 - 2x^{2}\right)\mathrm{e}^{1-x^{2}}$$ b. Deduce the table of variations of the function $f$.

A manufacturer must create a solid wooden gate made to measure for a homeowner. The opening of the enclosure wall (not yet built) cannot exceed 4 meters wide. The gate consists of two panels of width $a$ such that $0 < a \leqslant 2$.
In the chosen model, the closed gate has the shape illustrated in the figure. The sides $[\mathrm{AD}]$ and $[\mathrm{BC}]$ are perpendicular to the threshold [CD] of the gate. Between points A and B, the top of the panels has the shape of a portion of curve. This portion of curve is part of the graph of the function $f$ defined on $[-2 ; 2]$ by:
$$f ( x ) = - \frac { b } { 8 } \left( \mathrm { e } ^ { \frac { x } { b } } + \mathrm { e } ^ { - \frac { x } { b } } \right) + \frac { 9 } { 4 } \quad \text { where } b > 0 .$$
The coordinate system is chosen so that the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D have coordinates respectively $(-a ; f(-a))$, $(a ; f(a))$, $(a ; 0)$ and $(-a ; 0)$ and we denote S the vertex of the curve of $f$.
Part A

1. Show that, for all real $x$ belonging to the interval $[-2 ; 2], f(-x) = f(x)$. What can we deduce about the graph of the function $f$?
2. Let $f^{\prime}$ denote the derivative function of $f$. Show that, for all real $x$ in the interval $[-2 ; 2]$: $$f^{\prime}(x) = -\frac{1}{8}\left(\mathrm{e}^{\frac{x}{b}} - \mathrm{e}^{-\frac{x}{b}}\right)$$
3. Draw up the table of variations of the function $f$ on the interval $[-2 ; 2]$ and deduce the coordinates of point S as a function of $b$.

Part B
The height of the wall is $1.5\mathrm{~m}$. We want point S to be 2 m from the ground. We then seek the values of $a$ and $b$.

1. Justify that $b = 1$.
2. Show that the equation $f(x) = 1.5$ has a unique solution on the interval $[0 ; 2]$ and deduce an approximate value of $a$ to the nearest hundredth.
3. In this question, we choose $a = 1.8$ and $b = 1$. The customer decides to automate his gate if the mass of a panel exceeds 60 kg. The density of the wooden planks used to manufacture the panels is equal to $20\mathrm{~kg\cdot m^{-2}}$. What does the customer decide?

Part C
We keep the values $a = 1.8$ and $b = 1$. To cut the panels, the manufacturer pre-cuts planks. He has a choice between two forms of pre-cut planks: either a rectangle OCES, or a trapezoid OCHG. In the second method, the line (GH) is tangent to the graph of the function $f$ at point F with abscissa 1.
Form 1 is the simplest, but visually form 2 seems more economical. Evaluate the savings achieved in terms of wood surface area by choosing form 2 rather than form 1. We recall the formula giving the area of a trapezoid. By denoting $b$ and $B$ respectively the lengths of the small base and the large base of the trapezoid (parallel sides) and $h$ the height of the trapezoid: $$\text{Area} = \frac{b + B}{2} \times h$$

We consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = \frac { ( \ln x ) ^ { 2 } } { x }$$
We denote $\mathscr { C }$ the representative curve of $f$ in an orthonormal coordinate system.

1. Determine the limit of the function $f$ at 0 and interpret the result graphically.
2. a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$, $$f ( x ) = 4 \left( \frac { \ln ( \sqrt { x } ) } { \sqrt { x } } \right) ^ { 2 }$$ b. Deduce that the $x$-axis is an asymptote to the representative curve of the function $f$ in the neighbourhood of $+ \infty$.
3. We admit that $f$ is differentiable on $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function. a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$, $$f ^ { \prime } ( x ) = \frac { \ln ( x ) ( 2 - \ln ( x ) ) } { x ^ { 2 } } .$$ b. Study the sign of $f ^ { \prime } ( x )$ according to the values of the strictly positive real number $x$. c. Calculate $f ( 1 )$ and $f \left( \mathrm { e } ^ { 2 } \right)$.
4. Prove that the equation $f ( x ) = 1$ admits a unique solution $\alpha$ on $] 0 ; + \infty [$ and give a bound for $\alpha$ with amplitude $10 ^ { - 2 }$.

Exercise 1 (6 points) -- Part A
Let $a$ and $b$ be real numbers. We consider a function $f$ defined on $[ 0 ; + \infty [$ by $$f ( x ) = \frac { a } { 1 + \mathrm { e } ^ { - b x } }$$ The curve $\mathscr { C } _ { f }$ representing the function $f$ in an orthogonal coordinate system is given. The curve $\mathscr { C } _ { f }$ passes through the point $\mathrm { A } ( 0 ; 0.5 )$. The tangent line to the curve $\mathscr { C } _ { f }$ at point A passes through the point B(10; 1).

1. Justify that $a = 1$.

We then obtain, for all real $x \geqslant 0$, $$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - b x } }$$

1. It is admitted that the function $f$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function. Verify that, for all real $x \geqslant 0$ $$f ^ { \prime } ( x ) = \frac { b \mathrm { e } ^ { - b x } } { \left( 1 + \mathrm { e } ^ { - b x } \right) ^ { 2 } }$$
2. Using the data from the problem statement, determine $b$.

Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
Consider the function $f$ defined on the interval $]0; 1]$ by $$f(x) = x(1 - \ln x)^2.$$
a. Determine an expression for the derivative of $f$ and verify that for all $x \in ]0; 1]$, $f'(x) = (\ln x + 1)(\ln x - 1)$.
b. Study the variations of the function $f$ and draw its variation table on the interval $]0; 1]$ (it will be admitted that the limit of the function $f$ at 0 is zero).

In the plane with a coordinate system, we consider the curve $\mathscr{C}_{f}$ representative of a function $f$, twice differentiable on the interval $]0 ; +\infty[$. The curve $\mathscr{C}_{f}$ admits a horizontal tangent line $T$ at point A(1;4).

1. Specify the values $f(1)$ and $f^{\prime}(1)$.

We admit that the function $f$ is defined for every real number $x$ in the interval $]0 ; +\infty[$ by:
$$f(x) = \frac{a + b \ln x}{x} \text{ where } a \text{ and } b \text{ are two real numbers.}$$

1. Prove that, for every strictly positive real number $x$, we have: $$f^{\prime}(x) = \frac{b - a - b \ln x}{x^{2}}$$
2. Deduce the values of the real numbers $a$ and $b$.

In the rest of the exercise, we admit that the function $f$ is defined for every real number $x$ in the interval $]0; +\infty[$ by:
$$f(x) = \frac{4 + 4 \ln x}{x}$$

1. Determine the limits of $f$ at 0 and at $+\infty$.
2. Determine the variation table of $f$ on the interval $]0 ; +\infty[$.
3. Prove that, for every strictly positive real number $x$, we have: $$f^{\prime\prime}(x) = \frac{-4 + 8 \ln x}{x^{3}}$$
4. Show that the curve $\mathscr{C}_{f}$ has a unique inflection point B whose coordinates you will specify.

EXERCISE-A

Main topics covered: convexity, logarithm function

Part I: graphical readings

$f$ denotes a function defined and differentiable on $\mathbb{R}$. We give below the representative curve of the derivative function $f'$.

With the precision allowed by the graph, answer the following questions

1. Determine the slope of the tangent line to the curve of function $f$ at 0.
2. a. Give the variations of the derivative function $f'$. b. Deduce an interval on which $f$ is convex.

Part II: function study

The function $f$ is defined on $\mathbb{R}$ by $$f(x) = \ln\left(x^{2} + x + \frac{5}{2}\right)$$

1. Calculate the limits of function $f$ at $+\infty$ and at $-\infty$.
2. Determine an expression $f'(x)$ of the derivative function of $f$ for all $x \in \mathbb{R}$.
3. Deduce the table of variations of $f$. Be sure to place the limits in this table.
4. a. Justify that the equation $f(x) = 2$ has a unique solution $\alpha$ in the interval $\left[-\frac{1}{2}; +\infty\right[$. b. Give an approximate value of $\alpha$ to $10^{-1}$ near.
5. The function $f'$ is differentiable on $\mathbb{R}$. We admit that, for all $x \in \mathbb{R}$, $f''(x) = \frac{-2x^{2} - 2x + 4}{\left(x^{2} + x + \frac{5}{2}\right)^{2}}$. Determine the number of inflection points of the representative curve of $f$.

Main topics covered: Logarithm function; differentiation

Part 1

Let $h$ denote the function defined on the interval $]0; +\infty[$ by: $$h(x) = 1 + \frac{\ln(x)}{x^2}.$$ It is admitted that the function $h$ is differentiable on $]0; +\infty[$ and we denote $h'$ its derivative function.

1. Determine the limits of $h$ at 0 and at $+\infty$.
2. Show that, for every real number $x$ in $]0; +\infty[$, $h'(x) = \frac{1 - 2\ln(x)}{x^3}$.
3. Deduce the variations of the function $h$ on the interval $]0; +\infty[$.
4. Show that the equation $h(x) = 0$ admits a unique solution $\alpha$ belonging to $]0; +\infty[$ and verify that: $\frac{1}{2} < \alpha < 1$.
5. Determine the sign of $h(x)$ for $x$ belonging to $]0; +\infty[$.

Part 2

Let $f_1$ and $f_2$ denote the functions defined on $]0; +\infty[$ by: $$f_1(x) = x - 1 - \frac{\ln(x)}{x^2} \quad \text{and} \quad f_2(x) = x - 2 - \frac{2\ln(x)}{x^2}.$$ We denote $\mathscr{C}_1$ and $\mathscr{C}_2$ the respective graphs of $f_1$ and $f_2$ in a reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

1. Show that, for every real number $x$ belonging to $]0; +\infty[$, we have: $$f_1(x) - f_2(x) = h(x).$$
2. Deduce from the results of Part 1 the relative position of the curves $\mathscr{C}_1$ and $\mathscr{C}_2$.

You will justify that their unique point of intersection has coordinates $(\alpha; \alpha)$. Recall that $\alpha$ is the unique solution of the equation $h(x) = 0$.

Main topics covered: Exponential function; differentiation; convexity

Part 1

Below is given, in the plane referred to an orthonormal reference frame, the curve representing the derivative function $f'$ of a function $f$ differentiable on $\mathbb{R}$. Using this curve, conjecture, by justifying the answers:

1. The direction of variation of the function $f$ on $\mathbb{R}$.
2. The convexity of the function $f$ on $\mathbb{R}$.

Part 2

It is admitted that the function $f$ mentioned in Part 1 is defined on $\mathbb{R}$ by: $$f(x) = (x+2)\mathrm{e}^{-x}.$$ We denote $\mathscr{C}$ the representative curve of $f$ in an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. It is admitted that the function $f$ is twice differentiable on $\mathbb{R}$, and we denote $f'$ and $f''$ the first and second derivative functions of $f$ respectively.

1. Show that, for every real number $x$, $$f(x) = \frac{x}{\mathrm{e}^x} + 2\mathrm{e}^{-x}.$$ Deduce the limit of $f$ at $+\infty$. Justify that the curve $\mathscr{C}$ admits an asymptote which you will specify. It is admitted that $\lim_{x \rightarrow -\infty} f(x) = -\infty$.
2. a. Show that, for every real number $x$, $f'(x) = (-x-1)\mathrm{e}^{-x}$. b. Study the variations on $\mathbb{R}$ of the function $f$ and draw up its variation table. c. Show that the equation $f(x) = 2$ admits a unique solution $\alpha$ on the interval $[-2;-1]$ and give an approximate value to the nearest $10^{-1}$.
3. Determine, for every real number $x$, the expression of $f''(x)$ and study the convexity of the function $f$.

What does point A with abscissa 0 represent for the curve $\mathscr{C}$?

Part I

We consider the function $h$ defined on the interval $] 0 ; + \infty [$ by: $$h ( x ) = 1 + \frac { \ln ( x ) } { x }$$

1. Determine the limit of the function $h$ at 0.
2. Determine the limit of the function $h$ at $+ \infty$.
3. We denote $h ^ { \prime }$ the derivative function of $h$. Prove that, for every real number $x$ in $] 0 ; + \infty [$, we have: $$h ^ { \prime } ( x ) = \frac { 1 - \ln ( x ) } { x ^ { 2 } }$$
4. Draw up the variation table of the function $h$ on the interval $] 0 ; + \infty [$.
5. Prove that the equation $h ( x ) = 0$ has a unique solution $\alpha$ in $] 0 ; + \infty [$. Justify that we have: $0.5 < \alpha < 0.6$.

Part II

In this part, we consider the functions $f$ and $g$ defined on $] 0 ; + \infty [$ by: $$f ( x ) = x \ln ( x ) - x ; \quad g ( x ) = \ln ( x )$$ We denote $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ the curves representing respectively the functions $f$ and $g$ in an orthonormal coordinate system $(O ; \vec { \imath } , \vec { \jmath })$. For every strictly positive real number $a$, we call:

• $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at its point with abscissa $a$;
• $D _ { a }$ the tangent to $\mathscr { C } _ { g }$ at its point with abscissa $a$.

We are looking for possible values of $a$ for which the lines $T _ { a }$ and $D _ { a }$ are perpendicular. Let $a$ be a real number belonging to the interval $] 0 ; + \infty [$.

1. Justify that the line $D _ { a }$ has slope $\frac { 1 } { a }$.
2. Justify that the line $T _ { a }$ has slope $\ln ( a )$.
3. We recall that in an orthonormal coordinate system, two lines with slopes $m$ and $m ^ { \prime }$ respectively are perpendicular if and only if $m m ^ { \prime } = - 1$. Prove that there exists a unique value of $a$, which you will identify, for which the lines $T _ { a }$ and $D _ { a }$ are perpendicular.

Main topics covered: Logarithm function, limits, differentiation.

Part 1

The graph below gives the graphical representation in an orthonormal coordinate system of the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = \frac{2\ln(x) - 1}{x}$$

1. Determine by calculation the unique solution $\alpha$ of the equation $f(x) = 0$.
   Give the exact value of $\alpha$ as well as the value rounded to the nearest hundredth.
2. Specify, by graphical reading, the sign of $f(x)$ when $x$ varies in the interval $]0; +\infty[$.

Part II

We consider the function $g$ defined on the interval $]0; +\infty[$ by:
$$g(x) = [\ln(x)]^2 - \ln(x)$$

1. a. Determine the limit of the function $g$ at 0. b. Determine the limit of the function $g$ at $+\infty$.
2. We denote by $g'$ the derivative function of the function $g$ on the interval $]0; +\infty[$.
   Prove that, for any real number $x$ in $]0; +\infty[$, we have: $g'(x) = f(x)$, where $f$ denotes the function defined in Part I.
3. Draw up the variation table of the function $g$ on the interval $]0; +\infty[$.
   This table should include the limits of the function $g$ at 0 and at $+\infty$, as well as the value of the minimum of $g$ on $]0; +\infty[$.
4. Prove that, for any real number $m > -0.25$, the equation $g(x) = m$ has exactly two solutions.
5. Determine by calculation the two solutions of the equation $g(x) = 0$.

Main topics covered: Natural logarithm function; convexity
We consider the function $f$ defined on the interval $]0;+\infty[$ by: $$f(x) = x + 4 - 4\ln(x) - \frac{3}{x}$$ where ln denotes the natural logarithm function. We denote $\mathscr{C}$ the graphical representation of $f$ in an orthonormal coordinate system.

1. Determine the limit of the function $f$ at $+\infty$.
2. We assume that the function $f$ is differentiable on $]0;+\infty[$ and we denote $f^{\prime}$ its derivative function.
   Prove that, for every real number $x > 0$, we have: $$f^{\prime}(x) = \frac{x^{2} - 4x + 3}{x^{2}}$$
3. a. Give the variation table of the function $f$ on the interval $]0;+\infty[$.
   The exact values of the extrema and the limits of $f$ at 0 and at $+\infty$ will be shown. We will assume that $\lim_{x \rightarrow 0} f(x) = -\infty$. b. By simply reading the variation table, specify the number of solutions of the equation $f(x) = \frac{5}{3}$.
4. Study the convexity of the function $f$, that is, specify the parts of the interval $]0;+\infty[$ on which $f$ is convex, and those on which $f$ is concave. We will justify that the curve $\mathscr{C}$ admits a unique inflection point, whose coordinates we will specify.

Main topics covered: Exponential function; differentiation.
The graph below represents, in an orthogonal coordinate system, the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ defined on $\mathbb{R}$ by:
$$f(x) = x^{2}\mathrm{e}^{-x} \text{ and } g(x) = \mathrm{e}^{-x}.$$
Question 3 is independent of questions 1 and 2.

1. a. Determine the coordinates of the intersection points of $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$. b. Study the relative position of the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$.
2. For every real number $x$ in the interval $[-1; 1]$, we consider the points $M$ with coordinates $(x; f(x))$ and $N$ with coordinates $(x; g(x))$, and we denote by $d(x)$ the distance $MN$. We assume that: $d(x) = \mathrm{e}^{-x} - x^{2}\mathrm{e}^{-x}$. We assume that the function $d$ is differentiable on the interval $[-1; 1]$ and we denote by $d^{\prime}$ its derivative function. a. Show that $d^{\prime}(x) = \mathrm{e}^{-x}\left(x^{2} - 2x - 1\right)$. b. Deduce the variations of the function $d$ on the interval $[-1; 1]$. c. Determine the common abscissa $x_{0}$ of the points $M_{0}$ and $N_{0}$ allowing to obtain a maximum distance $d(x_{0})$, and give an approximate value to 0.1 of the distance $M_{0}N_{0}$.
3. Let $\Delta$ be the line with equation $y = x + 2$. We consider the function $h$ differentiable on $\mathbb{R}$ and defined by: $h(x) = \mathrm{e}^{-x} - x - 2$. By studying the number of solutions of the equation $h(x) = 0$, determine the number of intersection points of the line $\Delta$ and the curve $\mathscr{C}_{g}$.

Main topics covered: Logarithm function; differentiation.
Part I: Study of an auxiliary function
Let $g$ be the function defined on $]0; +\infty[$ by:
$$g(x) = \ln(x) + 2x - 2.$$

1. Determine the limits of $g$ at $+\infty$ and 0.
2. Determine the direction of variation of the function $g$ on $]0; +\infty[$.
3. Prove that the equation $g(x) = 0$ admits a unique solution $\alpha$ on $]0; +\infty[$.
4. Calculate $g(1)$ then determine the sign of $g$ on $]0; +\infty[$.

Part II: Study of a function $f$
We consider the function $f$, defined on $]0; +\infty[$ by:
$$f(x) = \left(2 - \frac{1}{x}\right)(\ln(x) - 1)$$

1. a. We assume that the function $f$ is differentiable on $]0; +\infty[$ and we denote by $f^{\prime}$ its derivative. Prove that, for every $x$ in $]0; +\infty[$, we have: $$f^{\prime}(x) = \frac{g(x)}{x^{2}}$$ b. Draw the variation table of the function $f$ on $]0; +\infty[$. The calculation of limits is not required.
2. Solve the equation $f(x) = 0$ on $]0; +\infty[$ then draw the sign table of $f$ on the interval $]0; +\infty[$.

Part III: Study of a function $F$ whose derivative is the function $f$
We assume that there exists a function $F$ differentiable on $]0; +\infty[$ whose derivative $F^{\prime}$ is the function $f$. Thus, we have: $F^{\prime} = f$. We denote by $\mathscr{C}_{F}$ the representative curve of the function $F$ in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$. We will not seek to determine an expression for $F(x)$.

1. Study the variations of $F$ on $]0; +\infty[$.
2. Does the curve $\mathscr{C}_{F}$ representative of $F$ admit tangent lines parallel to the x-axis? Justify the answer.