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: $$\text{Area} = \frac{b + B}{2} \times h$$

Let $k$ be a strictly positive real number. Consider the functions $f _ { k }$ defined on $\mathbb { R }$ by: $$f _ { k } ( x ) = x + k \mathrm { e } ^ { - x } .$$ We denote by $\mathscr { C } _ { k }$ the representative curve of function $f _ { k }$ in a plane with an orthonormal coordinate system.
For every strictly positive real number $k$, the function $f _ { k }$ admits a minimum on $\mathbb { R }$. The value at which this minimum is attained is the abscissa of the point denoted $A _ { k }$ on the curve $\mathscr { C } _ { k }$. It would seem that, for every strictly positive real number $k$, the points $A _ { k }$ are collinear. Is this the case?

Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
It is admitted that, for all real $a$ in the interval $]0; 1]$, the area of triangle $\mathrm{O}N_aP_a$ in square units is given by $\mathscr{A}(a) = \frac{1}{2}a(1 - \ln a)^2$.
Using the previous questions, determine for which value of $a$ the area $\mathscr{A}(a)$ is maximum. Determine this maximum area.

Part B

In this part, $k$ denotes a strictly positive real number. We consider the function $f$ defined on $\mathbb { R }$ by $$f ( x ) = ( x - 1 ) \mathrm { e } ^ { - k x } + 1 .$$ We admit that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function. In the plane with an orthonormal coordinate system ( $\mathrm { O } ; \mathrm { I } , \mathrm { J }$ ), we denote $\mathscr { C } _ { f }$ the representative curve of the function $f$. The tangent line $T$ to the curve $\mathscr { C } _ { f }$ at point A with abscissa 1 intersects the ordinate axis at a point denoted B.

1. a. Prove that for all real $x$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { - k x } ( - k x + k + 1 )$$ b. Prove that the ordinate of point B is equal to $g ( k )$ where $g$ is the function defined in Part A, with $g ( x ) = 1 - \mathrm{e}^{-x}$.
2. Using Part A, prove that point B belongs to the segment [OJ].

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 $\mathscr{C}$?

Exercise 2 (7 points) Theme: functions, exponential function

Part A

Let $p$ be the function defined on the interval $[ - 3 ; 4 ]$ by: $$p ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x + 1$$

1. Determine the variations of the function $p$ on the interval $[ - 3 ; 4 ]$.
2. Justify that the equation $p ( x ) = 0$ admits in the interval $[-3;4]$ a unique solution which will be denoted $\alpha$.
3. Determine an approximate value of the real number $\alpha$ to the nearest tenth.
4. Give the sign table of the function $p$ on the interval $[ - 3 ; 4 ]$.

Part B

Let $f$ be the function defined on the interval $[ - 3 ; 4 ]$ by: $$f ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + x ^ { 2 } }$$ We denote by $\mathscr { C } _ { f }$ its representative curve in an orthogonal coordinate system.

1. a. Determine the derivative of the function $f$ on the interval $[ - 3 ; 4 ]$. b. Justify that the curve $\mathscr { C } _ { f }$ admits a horizontal tangent at the point with abscissa 1.
2. The designers of a water slide use the curve $\mathscr { C } _ { f }$ as the profile of a water slide. They estimate that the water slide provides good sensations if the profile has at least two inflection points. a. Based on the graph, does the water slide seem to provide good sensations? Argue. b. It is admitted that the function $f ^ { \prime \prime }$, the second derivative of the function $f$, has the following expression for every real $x$ in the interval $[ - 3 ; 4 ]$: $$f ^ { \prime \prime } ( x ) = \frac { p ( x ) ( x - 1 ) \mathrm { e } ^ { x } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } }$$ where $p$ is the function defined in Part A. Using the above expression for $f ^ { \prime \prime }$, answer the question: ``does the water slide provide good sensations?''. Justify.

A biologist has modeled the evolution of a bacterial population (in thousands of entities) by the function $f$ defined on $[0; +\infty[$ by
$$f(t) = e^3 - e^{-0.5t^2 + t + 2}$$
where $t$ denotes the time in hours since the beginning of the experiment. Based on this modeling, he proposes the three statements below. For each of them, indicate, by justifying, whether it is true or false.

• Statement 1: ``The population increases permanently''.
• Statement 2: ``In the long term, the population will exceed 21000 bacteria''.
• Statement 3: ``The bacterial population will have a count of 10000 on two occasions over time''.

PART A We define on the interval $]0;+\infty[$ the function $g$ by: $$g(x) = \frac{2}{x} - \frac{1}{x^2} + \ln x \text{ where ln denotes the natural logarithm function.}$$ We admit that the function $g$ is differentiable on $]0;+\infty[ = I$ and we denote by $g'$ its derivative function.

1. Show that for $x > 0$, the sign of $g'(x)$ is that of the quadratic trinomial $(x^2 - 2x + 2)$.
2. Deduce that the function $g$ is strictly increasing on $]0;+\infty[$.
3. Show that the equation $g(x) = 0$ admits a unique solution on the interval $[0{,}5; 1]$, which we will denote $\alpha$.
4. We are given the sign table of $g$ on the interval $]0;+\infty[ = I$:
   

   $x$	0		$\alpha$	$+\infty$
   $g(x)$	$-$	0	$+$

   
   Justify this sign table using the results obtained in the previous questions.

PART B We consider the function $f$ defined on the interval $]0;+\infty[ = I$ by: $$f(x) = \mathrm{e}^x \ln x.$$ We denote by $\mathscr{C}_f$ the representative curve of $f$ in an orthonormal coordinate system.

1. We admit that the function $f$ is twice differentiable on $]0;+\infty[$, we denote by $f'$ its derivative function, $f''$ its second derivative function and we admit that: for every real number $x > 0$, $f'(x) = \mathrm{e}^x\left(\frac{1}{x} + \ln x\right)$. Prove that, for every real number $x > 0$, we have: $f''(x) = \mathrm{e}^x\left(\frac{2}{x} - \frac{1}{x^2} + \ln x\right)$.
2. We may note that for every real $x > 0$, $f''(x) = \mathrm{e}^x \times g(x)$, where $g$ denotes the function studied in part A.
3. a. Draw the sign table of the function $f''$ on $]0;+\infty[$. Justify. b. Justify that the curve $\mathscr{C}_f$ admits a unique inflection point A. c. Study the convexity of the function $f$ on the interval $]0;+\infty[$. Justify.
4. a. Calculate the limits of $f$ at the boundaries of its domain of definition. b. Show that $f'(\alpha) = \frac{\mathrm{e}^\alpha}{\alpha^2}(1-\alpha)$. We recall that $\alpha$ is the unique solution of the equation $g(x) = 0$. c. Prove that $f'(\alpha) > 0$ and deduce the sign of $f'(x)$ for $x$ belonging to $]0;+\infty[$. d. Deduce the complete variation table of the function $f$ on $]0;+\infty[$.

Exercise 1

Part A

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

1. Determine the limits of the function $g$ at the boundaries of its domain.
2. It is admitted that the function $g$ is differentiable on the interval $] 0 ; + \infty [$. Study the variations of the function $g$ on the interval $] 0 ; + \infty [$.
3. a. Prove that there exists a unique strictly positive real number $\alpha$ such that $g ( \alpha ) = 0$. b. Determine an interval containing $\alpha$ with amplitude $10 ^ { - 2 }$.
4. Deduce the sign table of the function $g$ on the interval $] 0 ; + \infty [$.

Part B

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

1. a. Determine the limit of the function $f$ at 0. b. Interpret the result graphically.
2. Determine the limit of the function $f$ at $+ \infty$.
3. It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.

Show that for every strictly positive real number $x$, we have $f ^ { \prime } ( x ) = \frac { g ( x ) } { x ^ { 2 } }$.
4. Deduce the variations of the function $f$ on the interval $] 0 ; + \infty [$.

Part C

Study the relative position of the curve $\mathscr { C } _ { f }$ and the representative curve of the natural logarithm function on the interval $] 0 ; + \infty [$.

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.

Part A

Consider the function $f$ defined by :
$$f ( x ) = x - \ln ( 1 + x ) .$$

1. Justify that the function $f$ is defined on the interval $] - 1 ; + \infty [$.
2. We admit that the function $f$ is differentiable on $] - 1 ; + \infty [$.

Determine the expression of its derivative function $f ^ { \prime } ( x )$.
3. a. Deduce the direction of variation of the function $f$ on the interval $] - 1 ; + \infty [$. b. Deduce the sign of the function $f$ on the interval $] - 1 ; 0 [$.
4. a. Show that, for all $x$ belonging to the interval $] - 1 ; + \infty [$, we have :
$$f ( x ) = \ln \left( \frac { \mathrm { e } ^ { x } } { 1 + x } \right) .$$
b. Deduce the limit at $+ \infty$ of the function $f$.

Part B

Consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 10$ and, for all natural integer $n$,
$$u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right) .$$
We admit that the sequence ( $u _ { n }$ ) is well defined.

1. Give the value rounded to the nearest thousandth of $u _ { 1 }$.
2. Using question 3. a. of Part A , prove by induction that, for all natural integer $n$, we have $u _ { n } \geqslant 0$.
3. Prove that the sequence ( $u _ { n }$ ) is decreasing.
4. Deduce from the previous questions that the sequence ( $u _ { n }$ ) converges.
5. Determine the limit of the sequence $\left( u _ { n } \right)$.

Part A Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(x + \frac{1}{2}\right)e^{-x} + x.$$

1. Determine the limits of $f$ at $-\infty$ and at $+\infty$.
2. It is admitted that $f$ is twice differentiable on $\mathbb{R}$. a. Prove that, for all $x \in \mathbb{R}$, $$f''(x) = \left(x - \frac{3}{2}\right)e^{-x}.$$ b. Deduce the variations and the minimum of the function $f'$ on $\mathbb{R}$. c. Justify that for all $x \in \mathbb{R}$, $f'(x) > 0$. d. Deduce that the equation $f(x) = 0$ admits a unique solution on $\mathbb{R}$. e. Give a value rounded to $10^{-3}$ of this solution.

Part B Consider a function $h$, defined and differentiable on $\mathbb{R}$, having an expression of the form $$h(x) = (ax + b)e^{-x} + x \text{, where } a \text{ and } b \text{ are two real numbers.}$$ In an orthonormal coordinate system shown below are:

• the representative curve $\mathscr{C}_h$ of the function $h$;
• the points A and B with coordinates respectively $(-2; -2.5)$ and $(2; 3.5)$.

1. Conjecture, with the precision allowed by the graph, the abscissae of any inflection points of the representative curve of the function $h$.
2. Knowing that the function $h$ admits on $\mathbb{R}$ a second derivative with expression $$h''(x) = -\frac{3}{2}e^{-x} + xe^{-x}.$$ validate or not the previous conjecture.
3. Determine an equation of the line (AB).
4. Knowing that the line (AB) is tangent to the representative curve of the function $h$ at the point with abscissa 0, deduce the values of $a$ and $b$.

Let $k$ be a strictly positive real number. The purpose of this exercise is to determine the number of solutions of the equation
$$\ln ( x ) = k x$$
with parameter $k$.
1. Graphical conjectures: Based on the graph (showing the curve $y = \ln(x)$, the line $y = x$ and the line $y = 0{,}2x$), conjecture the number of solutions of the equation $\ln ( x ) = k x$ for $k = 1$ then for $k = 0{,}2$.
2. Study of the case $k = 1$:
We consider the function $f$, defined and differentiable on $] 0 ; + \infty [$, by:
$$f ( x ) = \ln ( x ) - x .$$
We denote $f ^ { \prime }$ the derivative function of the function $f$. a. Calculate $f ^ { \prime } ( x )$. b. Study the direction of variation of the function $f$ on $] 0 ; + \infty [$.
Draw the variation table of the function $f$ showing the exact value of the extrema if there are any. The limits at the boundaries of the domain of definition are not expected. c. Deduce the number of solutions of the equation $\ln ( x ) = x$.
3. Study of the general case: $k$ is a strictly positive real number. We consider the function $g$ defined on $] 0 ; + \infty [$ by:
$$g ( x ) = \ln ( x ) - k x .$$
We admit that the variation table of the function $g$ is as follows:

$x$	0	$\frac { 1 } { k }$	$+ \infty$
$g ( x )$	$\longrightarrow$	$g \left( \frac { 1 } { k } \right)$
	$- \infty$		$- \infty$

a. Give, as a function of the sign of $g \left( \frac { 1 } { k } \right)$, the number of solutions of the equation $g ( x ) = 0$. b. Calculate $g \left( \frac { 1 } { k } \right)$ as a function of the real number $k$. c. Show that $g \left( \frac { 1 } { k } \right) > 0$ is equivalent to $\ln ( k ) < - 1$. d. Determine the set of values of $k$ for which the equation $\ln ( x ) = k x$ has exactly two solutions. e. Give, according to the values of $k$, the number of solutions of the equation $\ln ( x ) = k x$.

In this part, we consider that the function $f$, defined and twice differentiable on $[0; +\infty[$, is defined by
$$f(x) = (4x - 2)\mathrm{e}^{-x + 1}.$$
We will denote respectively $f'$ and $f''$ the derivative and second derivative of the function $f$.

1. Study of the function $f$ a. Show that $f'(x) = (-4x + 6)\mathrm{e}^{-x + 1}$. b. Use this result to determine the complete table of variations of the function $f$ on $[0; +\infty[$. It is admitted that $\lim_{x \rightarrow +\infty} f(x) = 0$. c. Study the convexity of the function $f$ and specify the abscissa of any possible inflection point of the representative curve of $f$.
2. We consider a function $F$ defined on $[0; +\infty[$ by $F(x) = (ax + b)\mathrm{e}^{-x + 1}$, where $a$ and $b$ are two real numbers. a. Determine the values of the real numbers $a$ and $b$ such that the function $F$ is a primitive of the function $f$ on $[0; +\infty[$. b. It is admitted that $F(x) = (-4x - 2)\mathrm{e}^{-x + 1}$ is a primitive of the function $f$ on $[0; +\infty[$. Deduce the exact value, then an approximate value to $10^{-2}$ near, of the integral $$I = \int_{\frac{3}{2}}^{8} f(x)\mathrm{d}x.$$
3. A municipality has decided to build a freestyle scooter track. The profile of this track is given by the representative curve of the function $f$ on the interval $[\frac{3}{2}; 8]$. The unit of length is the meter. a. Give an approximate value to the nearest cm of the height of the starting point D. b. The municipality has organized a graffiti competition to decorate the wall profile of the track. The selected artist plans to cover approximately $75\%$ of the wall surface. Knowing that a 150 mL aerosol can covers a surface of $0.8\mathrm{~m}^2$, determine the number of cans she will need to use to create this work.

Consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = x ^ { 2 } - x \ln ( x ) .$$
We admit that $f$ is twice differentiable on $] 0 ; + \infty [$. We denote $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the derivative function of $f ^ { \prime }$.
Part A: Study of the function $f$

1. Determine the limits of the function $f$ at 0 and at $+ \infty$.
2. For all strictly positive real $x$, calculate $f ^ { \prime } ( x )$.
3. Show that for all strictly positive real $x$: $$f ^ { \prime \prime } ( x ) = \frac { 2 x - 1 } { x }$$
4. Study the variations of the function $f ^ { \prime }$ on $] 0 ; + \infty [$, then draw up the table of variations of the function $f ^ { \prime }$ on $] 0 ; + \infty [$. Care should be taken to show the exact value of the extremum of the function $f ^ { \prime }$ on $] 0 ; + \infty [$. The limits of the function $f ^ { \prime }$ at the boundaries of the domain of definition are not expected.
5. Show that the function $f$ is strictly increasing on $] 0 ; + \infty [$.

Part B: Study of an auxiliary function for solving the equation $f ( x ) = x$
We consider in this part the function $g$ defined on $] 0 ; + \infty [$ by
$$g ( x ) = x - \ln ( x )$$
We admit that the function $g$ is differentiable on $] 0 ; + \infty [$, we denote $g ^ { \prime }$ its derivative.

1. For all strictly positive real, calculate $g ^ { \prime } ( x )$, then draw up the table of variations of the function $g$. The limits of the function $g$ at the boundaries of the domain of definition are not expected.
2. We admit that 1 is the unique solution of the equation $g ( x ) = 1$. Solve, on the interval $] 0 ; + \infty [$, the equation $f ( x ) = x$.

Part C: Study of a recursive sequence
We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = \frac { 1 } { 2 }$ and for all natural integer $n$,
$$u _ { n + 1 } = f \left( u _ { n } \right) = u _ { n } ^ { 2 } - u _ { n } \ln \left( u _ { n } \right) .$$

1. Show by induction that for all natural integer $n$: $$\frac { 1 } { 2 } \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 1 .$$
2. Justify that the sequence $( u _ { n } )$ converges. We call $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ satisfies the equality $f ( \ell ) = \ell$.
3. Determine the value of $\ell$.

Consider a function $f$ defined and twice differentiable on $]-2; +\infty[$. We denote by $\mathscr{C}_f$ its representative curve in an orthogonal coordinate system of the plane, $f'$ its derivative and $f''$ its second derivative. The curve $\mathscr{C}_f$ and its tangent $T$ at point B with abscissa $-1$ are drawn. It is specified that the line $T$ passes through the point $\mathrm{A}(0; -1)$.
Part A: exploitation of the graph.
Using the graph, answer the questions below.

1. Specify $f(-1)$ and $f'(-1)$.
2. Is the function $f$ convex on its domain of definition? Justify.
3. Conjecture the number of solutions of the equation $f(x) = 0$ and give a value rounded to $10^{-1}$ near a solution.

Part B: study of the function $f$
Consider that the function $f$ is defined on $]-2; +\infty[$ by: $$f(x) = x^2 + 2x - 1 + \ln(x+2),$$ where ln denotes the natural logarithm function.

1. Determine by calculation the limit of the function $f$ at $-2$. Interpret this result graphically.

It is admitted that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.

1. Show that for all $x > -2$, $\quad f'(x) = \frac{2x^2 + 6x + 5}{x+2}$.
2. Study the variations of the function $f$ on $]-2; +\infty[$ then draw up its complete variation table.
3. Show that the equation $f(x) = 0$ admits a unique solution $\alpha$ on $]-2; +\infty[$ and give a value of $\alpha$ rounded to $10^{-2}$ near.
4. Deduce the sign of $f(x)$ on $]-2; +\infty[$.
5. Show that $\mathscr{C}_f$ admits a unique inflection point and determine its abscissa.

Part C: a minimum distance.
Let $g$ be the function defined on $]-2; +\infty[$ by $\quad g(x) = \ln(x+2)$. We denote by $\mathscr{C}_g$ its representative curve in an orthonormal coordinate system $(O; I, J)$. Let $M$ be a point of $\mathscr{C}_g$ with abscissa $x$. The purpose of this part is to determine for which value of $x$ the distance $JM$ is minimal. Consider the function $h$ defined on $]-2; +\infty[$ by $\quad h(x) = JM^2$.

1. Justify that for all $x > -2$, we have: $\quad h(x) = x^2 + [\ln(x+2) - 1]^2$.
2. It is admitted that the function $h$ is differentiable on $]-2; +\infty[$ and we denote by $h'$ its derivative function. It is also admitted that for all real $x > -2$, $$h'(x) = \frac{2f(x)}{x+2}$$ where $f$ is the function studied in part B. a. Draw up the variation table of $h$ on $]-2; +\infty[$. The limits are not required. b. Deduce that the value of $x$ for which the distance $JM$ is minimal is $\alpha$ where $\alpha$ is the real number defined in question 4 of part B.
3. We will denote by $M_\alpha$ the point of $\mathscr{C}_g$ with abscissa $\alpha$. a. Show that $\ln(\alpha + 2) = 1 - 2\alpha - \alpha^2$. b. Deduce that the tangent to $\mathscr{C}_g$ at point $M_\alpha$ and the line $(JM_\alpha)$ are perpendicular. One may use the fact that, in an orthonormal coordinate system, two lines are perpendicular when the product of their slopes is equal to $-1$.

We propose to study in this part the function $f$ encountered in Part B question 2. We recall that, for every real $x$, $f(x) = \left(6x^2 + 2x - 2\right)\mathrm{e}^{-5x+1}$. We denote $f'$ the derivative function of the function $f$. We call $\mathscr{C}_f$ the representative curve of $f$ in a coordinate system of the plane.

1. We admit that $\lim_{x \rightarrow +\infty} f(x) = 0$. Determine the limit of the function $f$ at $-\infty$.
2. Using Part A (Exercise 4), show that $\mathscr{C}_f$ intersects the $x$-axis at two points (the coordinates of these points are not expected).
3. Using Parts A and B (Exercise 4), show that $\mathscr{C}_f$ has two horizontal tangent lines.
4. Draw the complete variation table of the function $f$.
5. Determine by justifying the number of solution(s) of the equation $f(x) = 1$.
6. For every real $m$ strictly greater than 0.2, we define $I_m$ by $I_m = \int_{0.2}^{m} f(x)\,\mathrm{d}x$. a. Verify that the function $F$ defined on $\mathbb{R}$ by $$F(x) = \left(-\frac{6}{5}x^2 - \frac{22}{25}x + \frac{28}{125}\right)\mathrm{e}^{-5x+1}$$ is a primitive of the function $f$ on $\mathbb{R}$. b. Does there exist a value of $m$ for which $I_m = 0$? Interpret this result graphically.

To prevent an epidemic, the Health Department of a city disinfected all neighborhoods to prevent the spread of the dengue mosquito. It is known that the number $f$ of infected people is given by the function $f(t) = -2t^{2} + 120t$ (where $t$ is expressed in days and $t = 0$ is the day before the first infection) and that this expression is valid for the first 60 days of the epidemic.
The Health Department decided that a second disinfection should be done on the day when the number of infected people reached 1600 people, and a second disinfection had to take place.
The second disinfection began on
(A) the $19^{\text{th}}$ day.
(B) the $20^{\text{th}}$ day.
(C) the $29^{\text{th}}$ day.
(D) the $30^{\text{th}}$ day.
(E) the $60^{\text{th}}$ day.

Having a large piece of land, an entertainment company intends to build a rectangular space for shows and events, as shown in the figure.
The area for the public will be fenced with two types of materials:

• on the sides parallel to the stage, a type A screen will be used, more resistant, whose value per linear meter is $\mathrm{R}\$ 20.00$;
• on the other two sides, a type B screen will be used, common, whose linear meter costs $\mathrm{R}\$ 5.00$.

The company has $\mathrm{R}\$ 5000.00$ to buy all the screens, but wants to do it in such a way that it obtains the largest possible area for the public.
The quantity of each type of screen that the company should buy is
(A) $50.0 \mathrm{~m}$ of type A screen and $800.0 \mathrm{~m}$ of type B screen.
(B) $62.5 \mathrm{~m}$ of type A screen and $250.0 \mathrm{~m}$ of type B screen.
(C) $100.0 \mathrm{~m}$ of type A screen and $600.0 \mathrm{~m}$ of type B screen.
(D) $125.0 \mathrm{~m}$ of type A screen and $500.0 \mathrm{~m}$ of type B screen.
(E) $200.0 \mathrm{~m}$ of type A screen and $200.0 \mathrm{~m}$ of type B screen.

Lobster hatcheries are built, by local fishing cooperatives, in the shape of right-rectangular prisms, fixed to the ground and with flexible nets of the same height, capable of withstanding marine corrosion. For each hatchery to be built, the cooperative uses entirely 100 linear meters of this net, which is used only on the sides.
What should be the values of $X$ and $Y$, in meters, so that the area of the base of the hatchery is maximum?
(A) 1 and 49
(B) 1 and 99
(C) 10 and 10
(D) 25 and 25
(E) 50 and 50

$f ( x ) = x ^ { 3 } + x ^ { 2 } + c x + d$, where $c$ and $d$ are real numbers. Prove that if $c > \frac { 1 } { 3 }$, then $f$ has exactly one real root.

Consider the function $f ( x ) = a x + \frac { 1 } { x + 1 }$, where $a$ is a positive constant. Let $L =$ the largest value of $f ( x )$ and $S =$ the smallest value of $f ( x )$ for $x \in [ 0,1 ]$. Show that $L - S > \frac { 1 } { 12 }$ for any $a > 0$.

Let $f ( x ) = ( x - a ) ( x - b ) ^ { 3 } ( x - c ) ^ { 5 } ( x - d ) ^ { 7 }$, where $a , b , c , d$ are real numbers with $a < b < c < d$. Thus $f ( x )$ has 16 real roots counting multiplicities and among them 4 are distinct from each other. Consider $f ^ { \prime } ( x )$, i.e. the derivative of $f ( x )$. Find the following, if you can: (i) the number of real roots of $f ^ { \prime } ( x )$, counting multiplicities, (ii) the number of distinct real roots of $f ^ { \prime } ( x )$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function, where $\mathbb{R}$ denotes the set of real numbers. Suppose that for all real numbers $x$ and $y$, the function $f$ satisfies
$$f'(x) - f'(y) \leq 3|x - y|$$
Answer the following questions. No credit will be given without full justification.
(a) Show that for all $x$ and $y$, we must have $\left|f(x) - f(y) - f'(y)(x - y)\right| \leq 1.5(x - y)^2$.
(b) Find the largest and smallest possible values for $f''(x)$ under the given conditions.

Let $g$ be a function such that all its derivatives exist. We say $g$ has an inflection point at $x_0$ if the second derivative $g''$ changes sign at $x_0$ i.e., if $g''(x_0 - \epsilon) \times g''(x_0 + \epsilon) < 0$ for all small enough positive $\epsilon$.
(a) If $g''(x_0) = 0$ then $g$ has an inflection point at $x_0$. True or False?
(b) If $g$ has an inflection point at $x_0$ then $g''(x_0) = 0$. True or False?
(c) Find all values $x_0$ at which $x^{4}(x - 10)$ has an inflection point.