Let $f$ be the differentiable function defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \mathrm { e } ^ { x } + \frac { 1 } { x }$$
1. Study of an auxiliary function
a. Let the function $g$ be differentiable, defined on $[ 0 ; + \infty [$ by
$$g ( x ) = x ^ { 2 } \mathrm { e } ^ { x } - 1 .$$
Study the direction of variation of the function $g$.
b. Prove that there exists a unique real number $a$ belonging to $[ 0 ; + \infty [$ such that $g ( a ) = 0$.
Prove that $a$ belongs to the interval $[ 0{,}703 ; 0{,}704 [$.
c. Determine the sign of $g ( x )$ on $[ 0 ; + \infty [$.
2. Study of the function $f$
a. Determine the limits of the function $f$ at 0 and at $+ \infty$.
b. Let $f ^ { \prime }$ denote the derivative function of $f$ on the interval $] 0 ; + \infty [$.
Prove that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { g ( x ) } { x ^ { 2 } }$.
c. Deduce the direction of variation of the function $f$ and draw its variation table on the interval $] 0 ; + \infty [$.
d. Prove that the function $f$ has as its minimum the real number
$$m = \frac { 1 } { a ^ { 2 } } + \frac { 1 } { a } .$$
e. Justify that $3{,}43 < m < 3{,}45$.

Consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = ( x + 2 ) \mathrm { e } ^ { - x }.$$
We denote by $\mathscr { C }$ the representative curve of the function $f$ in an orthogonal coordinate system.

1. Study of the function $f$. a. Determine the coordinates of the intersection points of the curve $\mathscr { C }$ with the axes of the coordinate system. b. Study the limits of the function $f$ at $- \infty$ and at $+ \infty$. Deduce any possible asymptotes of the curve $\mathscr { C }$. c. Study the variations of $f$ on $\mathbb { R }$.
2. Calculation of an approximate value of the area under a curve.

We denote by $\mathscr { D }$ the region between the $x$-axis, the curve $\mathscr { C }$ and the lines with equations $x = 0$ and $x = 1$. We approximate the area of the region $\mathscr { D }$ by calculating a sum of areas of rectangles. a. In this question, we divide the interval $[ 0 ; 1 ]$ into four intervals of equal length:

• On the interval $\left[ 0 ; \frac { 1 } { 4 } \right]$, we construct a rectangle of height $f ( 0 )$
• On the interval $\left[ \frac { 1 } { 4 } ; \frac { 1 } { 2 } \right]$, we construct a rectangle of height $f \left( \frac { 1 } { 4 } \right)$
• On the interval $\left[ \frac { 1 } { 2 } ; \frac { 3 } { 4 } \right]$, we construct a rectangle of height $f \left( \frac { 1 } { 2 } \right)$
• On the interval $\left[ \frac { 3 } { 4 } ; 1 \right]$, we construct a rectangle of height $f \left( \frac { 3 } { 4 } \right)$

The algorithm below allows us to obtain an approximate value of the area of the region $\mathscr { D }$ by adding the areas of the four preceding rectangles:

Variables :	$k$ is an integer
	$S$ is a real number
Initialization :	Assign to $S$ the value 0
Processing:	For $k$ varying from 0 to 3
	$\mid$ Assign to $S$ the value $S + \frac { 1 } { 4 } f \left( \frac { k } { 4 } \right)$
	End For
Output :	Display $S$

Give an approximate value to $10 ^ { - 3 }$ of the result displayed by this algorithm. b. In this question, $N$ is an integer strictly greater than 1. We divide the interval $[ 0 ; 1 ]$ into $N$ intervals of equal length. On each of these intervals, we construct a rectangle by proceeding in the same manner as in question 2.a. Modify the preceding algorithm so that it displays as output the sum of the areas of the $N$ rectangles thus constructed.
3. Calculation of the exact value of the area under a curve.
Let $g$ be the function defined on $\mathbb { R }$ by
$$g ( x ) = ( - x - 3 ) \mathrm { e } ^ { - x }$$
We admit that $g$ is an antiderivative of the function $f$ on $\mathbb { R }$. a. Calculate the area $\mathscr { A }$ of the region $\mathscr { D }$, expressed in square units. b. Give an approximate value to $10 ^ { - 3 }$ of the error made by replacing $\mathscr { A }$ by the approximate value found using the algorithm of question 2.a, that is the difference between these two values.

Let $f$ be the function defined on the interval $]0; +\infty[$ by: $$f(x) = x - 5\ln x - \frac{4}{x}$$

1. Determine the limit of $f(x)$ as $x$ tends to 0. You may use without proof the fact that $\lim_{x \rightarrow 0} x \ln x = 0$.
2. Determine the limit of $f(x)$ as $x$ tends to $+\infty$.
3. Prove that, for all strictly positive real $x$, $f'(x) = u(x)$, where $u(x) = \frac{x^2 - 5x + 4}{x^2}$.

Deduce the table of variations of the function $f$ by specifying the limits and particular values.

Consider the function $f$ defined and differentiable on the interval $[0 ; +\infty[$ by
$$f ( x ) = x \mathrm { e } ^ { - x } - 0,1$$

1. Determine the limit of $f$ as $x \to +\infty$.
2. Study the variations of $f$ on $[0 ; +\infty[$ and draw the variation table.
3. Prove that the equation $f ( x ) = 0$ has a unique solution denoted $\alpha$ on the interval $[0 ; 1]$.

We admit the existence of a strictly positive real number $\beta$ such that $\alpha < \beta$ and $f ( \beta ) = 0$. We denote by $\mathscr { C }$ the representative curve of the function $f$ on the interval $[\alpha ; \beta]$ in an orthogonal coordinate system and $\mathscr { C } ^ { \prime }$ the curve symmetric to $\mathscr { C }$ with respect to the $x$-axis.
The unit on each axis represents 5 meters. These curves are used to delimit a floral bed in the shape of a candle flame on which tulips will be planted.

1. Prove that the function $F$, defined on the interval $[\alpha ; \beta]$ by $$F ( x ) = - ( x + 1 ) \mathrm { e } ^ { - x } - 0,1 x$$ is an antiderivative of the function $f$ on the interval $[\alpha ; \beta]$.
2. Calculate, in square units, a value rounded to 0.01 of the area of the region between the curves $\mathscr { C }$ and $\mathscr { C } ^ { \prime }$. Use the following values rounded to 0.001: $\alpha \approx 0.112$ and $\beta \approx 3.577$.
3. Knowing that 36 tulip plants can be placed per square meter, calculate the number of tulip plants needed for this floral bed.

A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
Part B: study of functions

1. Let $f$ be the function defined on $]0; +\infty[$ by:
   $$f ( x ) = \frac { 105 } { x } \left( 1 - \mathrm { e } ^ { - \frac { 3 } { 40 } x } \right)$$
   Prove that, for every real $x$ in $]0; +\infty[$, $f ^ { \prime } ( x ) = \frac { 105 g ( x ) } { x ^ { 2 } }$, where $g$ is the function defined on $[0; +\infty[$ by:
   $$g ( x ) = \frac { 3 x } { 40 } \mathrm { e } ^ { - \frac { 3 } { 40 } x } + \mathrm { e } ^ { - \frac { 3 } { 40 } x } - 1$$
   
2. The variation table of the function $g$ is given:
   

   $x$	0	$+\infty$
   	0
   $g ( x )$		- 1

   
   Deduce the monotonicity of the function $f$. The limits of the function $f$ are not required.
   
3. Show that the equation $f ( x ) = 5.9$ has a unique solution on the interval $[1; 80]$. Deduce that this equation has a unique solution on the interval $]0; +\infty[$. Give an approximate value of this solution to the nearest tenth.

Let $f$ be the function defined on the interval $]0;+\infty[$ by: $$f(x) = \frac{\mathrm{e}^{x}}{x}.$$ We denote $\mathscr{C}_{f}$ the representative curve of the function $f$ in an orthonormal coordinate system.

1. a. Specify the limit of the function $f$ at $+\infty$. b. Justify that the $y$-axis is an asymptote to the curve $\mathscr{C}_{f}$.
2. Show that, for every real number $x$ in the interval $]0;+\infty[$, we have: $$f^{\prime}(x) = \frac{\mathrm{e}^{x}(x-1)}{x^{2}}$$ where $f^{\prime}$ denotes the derivative function of the function $f$.
3. Determine the variations of the function $f$ on the interval $]0;+\infty[$. A variation table of the function $f$ will be established in which the limits appear.
4. Let $m$ be a real number. Specify, depending on the values of the real number $m$, the number of solutions of the equation $f(x) = m$.
5. We denote $\Delta$ the line with equation $y = -x$.
   We denote A a possible point of $\mathscr{C}_{f}$ with abscissa $a$ at which the tangent to the curve $\mathscr{C}_{f}$ is parallel to the line $\Delta$. a. Show that $a$ is a solution of the equation $\mathrm{e}^{x}(x-1) + x^{2} = 0$.
   We denote $g$ the function defined on $[0;+\infty[$ by $g(x) = \mathrm{e}^{x}(x-1) + x^{2}$. We assume that the function $g$ is differentiable and we denote $g^{\prime}$ its derivative function. b. Calculate $g^{\prime}(x)$ for every real number $x$ in the interval $[0;+\infty[$, then establish the variation table of $g$ on $[0;+\infty[$. c. Show that there exists a unique point $A$ at which the tangent to $\mathscr{C}_{f}$ is parallel to the line $\Delta$.

Given the function
$$f ( x ) = x ^ { 3 } - 3 a x ^ { 2 } - 3 ( 2 a + 1 ) x + a + 2 ,$$
answer the following questions.
(1) For $\mathbf { G }$ $\sim$ $\mathbf { K }$, choose the correct answers from among (0) $\sim$ (5) below, and for the other $\square$, enter the correct numbers.
Since
$$f ^ { \prime } ( x ) = \mathbf { A } ( x - \mathbf { B } a - \mathbf { C } ) ( x + \mathbf { D } ) ,$$
we see that
(i) when $a > \mathbf { EF }$, $f ( x )$ is $\mathbf { G }$ at $x = - \square \mathbf { D }$ and is $\square$ H at $x =$ $\square$ B $a +$ $\square$ C;
(ii) when $a =$ $\square$ EF, $f ( x )$ is always $\square$ I;
(iii) when $a < \mathbf{EF}$, $f ( x )$ is $\square$ J at $x = -$ $\square$ D and is $\square$ K at $x =$ $\square$ B $a +$ $\square$ C. (0) locally maximized
(1) locally minimized
(2) increasing
(3) decreasing
(4) maximized
(5) minimized
(2) When we express the minimum value $m$ of $f ( x )$ over the range $- 1 \leqq x \leqq 1$ in terms of $a$, we have that
(i) when $a \geqq \mathbf { L }$, $m = \mathbf { MN } a$;
(ii) when $\mathbf { OP } \leqq a < \mathbf { L }$, $m = \mathbf { QR } \left( a ^ { 3 } + \mathbf { S } a ^ { 2 } + \mathbf { T } a \right)$;
(iii) when $a < \mathbf{OP}$, $m = \mathbf { U } a + \mathbf { V }$.
(3) The value of $m$ in (2) is maximized at $a = \frac { - \mathbf { W } + \sqrt { \mathbf { X } } } { \square \mathbf { Y } }$.