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.

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 $\Delta$.

Part I

We consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = x - \mathrm{e}^{-2x}$$
We call $\Gamma$ the representative curve of the function $f$ in an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath})$.

1. Determine the limits of the function $f$ at $-\infty$ and at $+\infty$.
2. Study the monotonicity of the function $f$ on $\mathbb{R}$ and draw up its variation table.
3. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on $\mathbb{R}$, and give an approximate value to $10^{-2}$ precision.
4. Deduce from the previous questions the sign of $f(x)$ according to the values of $x$.

Part II

In the orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath})$, we call $\mathscr{C}$ the representative curve of the function $g$ defined on $\mathbb{R}$ by:
$$g(x) = \mathrm{e}^{-x}$$
The curves $\mathscr{C}$ and the curve $\Gamma$ (which represents the function $f$ from Part I) are drawn on the graph provided in the appendix which is to be completed and returned with your paper. The purpose of this part is to determine the point on the curve $\mathscr{C}$ closest to the origin O of the coordinate system and to study the tangent to $\mathscr{C}$ at this point.

1. For any real number $t$, we denote by $M$ the point with coordinates $(t; \mathrm{e}^{-t})$ on the curve $\mathscr{C}$.
   We consider the function $h$ which, to the real number $t$, associates the distance $OM$. We therefore have: $h(t) = OM$, that is:
   $$h(t) = \sqrt{t^2 + \mathrm{e}^{-2t}}$$
   a. Show that, for any real number $t$,
   $$h'(t) = \frac{f(t)}{\sqrt{t^2 + \mathrm{e}^{-2t}}}$$
   where $f$ denotes the function studied in Part I. b. Prove that the point A with coordinates $(\alpha; \mathrm{e}^{-\alpha})$ is the point on the curve $\mathscr{C}$ for which the length $OM$ is minimal. Place this point on the graph provided in the appendix, to be returned with your paper.
   
2. We call $T$ the tangent to the curve $\mathscr{C}$ at A. a. Express in terms of $\alpha$ the slope of the tangent $T$.
   We recall that the slope of the line (OA) is equal to $\frac{\mathrm{e}^{-\alpha}}{\alpha}$. We also recall the following result which may be used without proof: In an orthonormal coordinate system of the plane, two lines $D$ and $D'$ with slopes $m$ and $m'$ respectively are perpendicular if and only if the product $mm'$ is equal to $-1$. b. Prove that the line (OA) and the tangent $T$ are perpendicular.
   Draw these lines on the graph provided in the appendix, to be returned with your paper.

Part A

Consider the function $f$ defined on the interval $[ 1 ; + \infty [$ by $$f ( x ) = \frac { \ln x } { x }$$ where ln denotes the natural logarithm function.

1. Give the limit of the function $f$ at $+ \infty$.
2. We admit that the function $f$ is differentiable on the interval $[ 1 ; + \infty [$ and we denote by $f ^ { \prime }$ its derivative function. a. Show that, for every real number $x \geqslant 1$, $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$. b. Justify the following sign table, giving the sign of $f ^ { \prime } ( x )$ according to the values of $x$.
   

   $x$	1		e	$+ \infty$
   $f ^ { \prime } ( x )$		+	0	-

   
   c. Draw up the complete variation table of the function $f$.
3. Let $k$ be a non-negative real number. a. Show that, if $0 \leqslant k \leqslant \frac { 1 } { \mathrm { e } }$, the equation $f ( x ) = k$ admits a unique solution on the interval $[1; e]$. b. If $k > \frac { 1 } { \mathrm { e } }$, does the equation $f ( x ) = k$ admit solutions on the interval $[ 1 ; + \infty [$? Justify.

Part B

Let $g$ be the function defined on $\mathbb { R }$ by: $$g ( x ) = \mathrm { e } ^ { \frac { x } { 4 } } .$$ We consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$: $u _ { n + 1 } = e ^ { \frac { u _ { n } } { 4 } }$, that is: $u _ { n + 1 } = g \left( u _ { n } \right)$.

1. Justify that the function $g$ is increasing on $\mathbb { R }$.
2. Show by induction that, for every natural integer $n$, we have: $u _ { n } \leqslant u _ { n + 1 } \leqslant \mathrm { e }$.
3. Deduce that the sequence $( u _ { n } )$ is convergent.

We denote by $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ is a solution of the equation: $$\mathrm { e } ^ { \frac { x } { 4 } } = x .$$

1. Deduce that $\ell$ is a solution of the equation $f ( x ) = \frac { 1 } { 4 }$, where $f$ is the function studied in Part A.
2. Give an approximate value to $10 ^ { - 2 }$ near of the limit $\ell$ of the sequence $( u _ { n } )$.

Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = x ^ { 2 } - 6 x + 4 \ln ( x )$$
It is admitted that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative. We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthogonal coordinate system.

1. a. Determine $\lim _ { x \rightarrow 0 } f ( x )$.
   Interpret this result graphically. b. Determine $\lim _ { x \rightarrow + \infty } f ( x )$.
2. a. Determine $f ^ { \prime } ( x )$ for all real $x$ belonging to $] 0 ; + \infty [$. b. Study the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$.
   Deduce the variation table of $f$.
3. Show that the equation $f ( x ) = 0$ has a unique solution in the interval $[4;5]$.
4. It is admitted that, for all $x$ in $] 0 ; + \infty [$, we have:
   $$f ^ { \prime \prime } ( x ) = \frac { 2 x ^ { 2 } - 4 } { x ^ { 2 } }$$
   a. Study the convexity of the function $f$ on $] 0 ; + \infty [$.
   The exact coordinates of any inflection points of $\mathscr { C } _ { f }$ will be specified. b. We denote A the point with coordinates $( \sqrt { 2 } ; f ( \sqrt { 2 } ) )$.
   Let $t$ be a strictly positive real number such that $t \neq \sqrt { 2 }$. Let $M$ be the point with coordinates $( t ; f ( t ) )$. Using question 4. a, indicate, according to the value of $t$, the relative positions of the segment [AM] and the curve $\mathscr { C } _ { f }$.

Exercise 1 — Multiple Choice (Exponential function)
For each of the following questions, only one of the four proposed answers is correct.

1. Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \frac{x}{\mathrm{e}^{x}}$$ We assume that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. a. $f'(x) = \mathrm{e}^{-x}$ b. $f'(x) = x\mathrm{e}^{-x}$ c. $f'(x) = (1-x)\mathrm{e}^{-x}$ d. $f'(x) = (1+x)\mathrm{e}^{-x}$
   
2. Let $f$ be a function twice differentiable on the interval $[-3;1]$. The graphical representation of its second derivative function $f''$ is given. We can then affirm that: a. The function $f$ is convex on the interval $[-2;0]$ b. The function $f$ is concave on the interval $[-1;1]$ c. The function $f'$ is decreasing on the interval $[-2;0]$ d. The function $f'$ admits a maximum at $x = -1$
   
3. We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = x^3 \mathrm{e}^{-x^2}$$ If $F$ is an antiderivative of $f$ on $\mathbb{R}$, a. $F(x) = -\frac{1}{6}\left(x^3+1\right)\mathrm{e}^{-x^2}$ b. $F(x) = -\frac{1}{4}x^4 \mathrm{e}^{-x^2}$ c. $F(x) = -\frac{1}{2}\left(x^2+1\right)\mathrm{e}^{-x^2}$ d. $F(x) = x^2\left(3-2x^2\right)\mathrm{e}^{-x^2}$
   
4. What is the value of: $$\lim_{x \rightarrow +\infty} \frac{\mathrm{e}^x + 1}{\mathrm{e}^x - 1}$$ a. $-1$ b. $1$ c. $+\infty$ d. does not exist
   
5. We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^{2x+1}$. The only antiderivative $F$ on $\mathbb{R}$ of the function $f$ such that $F(0) = 1$ is the function: a. $x \longmapsto 2\mathrm{e}^{2x+1} - 2\mathrm{e} + 1$ b. $x \longmapsto 2\mathrm{e}^{2x+1} - \mathrm{e}$ c. $x \longmapsto \frac{1}{2}\mathrm{e}^{2x+1} - \frac{1}{2}\mathrm{e} + 1$ d. $x \longmapsto \mathrm{e}^{x^2+x}$
   
6. In a coordinate system, the representative curve of a function $f$ defined and twice differentiable on $[-2;4]$ is drawn. Among the following curves (a, b, c, d), which one represents the function $f''$, the second derivative of $f$?

Exercise 3 (7 points)
Part 1
Let $g$ be the function defined for every real number $x$ in the interval $]0; +\infty[$ by: $$g(x) = \frac{2\ln x}{x}$$

1. Let $g'$ denote the derivative of $g$. Prove that for every strictly positive real $x$: $$g'(x) = \frac{2 - 2\ln x}{x^2}$$
2. We have the following variation table for the function $g$ on the interval $]0; +\infty[$:
   

   $x$	0	1	e	$+\infty$
   \begin{tabular}{ c } Variations
   of $g$

   & & & ${}^{\frac{2}{\mathrm{e}}}$ & & & & & & & & & \hline \end{tabular}
   Justify the following information read from this table: a. the value $\frac{2}{\mathrm{e}}$; b. the variations of the function $g$ on its domain; c. the limits of the function $g$ at the boundaries of its domain.
3. Deduce the sign table of the function $g$ on the interval $]0; +\infty[$.

Part 2
Let $f$ be the function defined on the interval $]0; +\infty[$ by $$f(x) = [\ln(x)]^2.$$ In this part, each study is carried out on the interval $]0; +\infty[$.

1. Prove that on the interval $]0; +\infty[$, the function $f$ is a primitive of the function $g$.
2. Using Part 1, study: a. the convexity of the function $f$; b. the variations of the function $f$.
3. a. Give an equation of the tangent line to the curve representing the function $f$ at the point with abscissa $e$. b. Deduce that, for every real $x$ in $]0; e]$: $$[\ln(x)]^2 \geqslant \frac{2}{\mathrm{e}} x - 1$$

We consider the function $f$ defined on $]0; +\infty[$ by $$f(x) = x^2 - 8\ln(x)$$ where ln denotes the natural logarithm function. We admit that $f$ is differentiable on $]0; +\infty[$, we denote by $f'$ its derivative function.

1. Determine $\lim_{x \rightarrow 0} f(x)$.
2. We admit that, for all $x > 0$, $f(x) = x^2\left(1 - 8\frac{\ln(x)}{x^2}\right)$.
   Deduce the limit: $\lim_{x \rightarrow +\infty} f(x)$.
3. Show that, for all real $x$ in $]0; +\infty[$, $f'(x) = \frac{2(x^2 - 4)}{x}$.
4. Study the variations of $f$ on $]0; +\infty[$ and draw up its complete variation table. We will specify the exact value of the minimum of $f$ on $]0; +\infty[$.
5. Prove that, on the interval $]0; 2]$, the equation $f(x) = 0$ admits a unique solution $\alpha$ (we will not seek to determine the value of $\alpha$).
6. We admit that, on the interval $[2; +\infty[$, the equation $f(x) = 0$ admits a unique solution $\beta$ (we will not seek to determine the value of $\beta$). Deduce the sign of $f$ on the interval $]0; +\infty[$.
7. For any real number $k$, we consider the function $g_k$ defined on $]0; +\infty[$ by: $$g_k(x) = x^2 - 8\ln(x) + k$$ Using the variation table of $f$, determine the smallest value of $k$ for which the function $g_k$ is positive on the interval $]0; +\infty[$.

We consider the function $f$ defined on the set $] 0 ; + \infty [$ by
$$f ( x ) = 1 + x ^ { 2 } - 2 x ^ { 2 } \ln ( x )$$
We admit that $f$ is differentiable on the interval and we denote $f ^ { \prime }$ its derivative function.

1. Justify that $\lim _ { x \rightarrow 0 } f ( x ) = 1$ and, by noting that $f ( x ) = 1 + x ^ { 2 } [ 1 - 2 \ln ( x ) ]$, justify that $\lim _ { x \rightarrow + \infty } f ( x ) = - \infty$.
2. Show that for all real $x$ in the interval $] 0 ; + \infty \left[ , f ^ { \prime } ( x ) = - 4 x \ln ( x ) \right.$.
3. Study the sign of $f ^ { \prime } ( x )$ on the interval $] 0$; $+ \infty [$, then draw up the table of variations of the function on the interval $] 0 ; + \infty [$.
4. Prove that the equation $f ( x ) = 0$ admits a unique solution $\alpha$ in the interval $[ 1 ; + \infty [$ and that $\alpha \in [ 1 ; \mathrm { e } ]$.

We admit in the rest of the exercise that the equation $f ( x ) = 0$ has no solution on the interval $] 0 ; 1]$.
5. We are given the function below written in Python. The instruction from lycee import* allows access to the function $\ln$.
\begin{verbatim} from lycee import * def f(x) : return 1 + x**2 - 2*x**2*ln(x) def dichotomie(p) a=1 b=2.7 while b - a > 10**(-p) : if f(a)*f((a+b)/2) < 0 : | b = (a+b)/2 else : |a =(a+b)/2 return (a,b) \end{verbatim}
It writes in the execution console:
\begin{verbatim} >>> dichotomie(1) \end{verbatim}
Among the four propositions below, copy the one displayed by the previous instruction. Justify your answer (you may proceed by elimination).
Proposition A: $\quad ( 1.75,1.9031250000000002 )$ Proposition B : ( $1.85,1.9031250000000002 )$ Proposition C : $\quad ( 2.75,2.9031250000000002 )$ Proposition D : (2.85, 2.9031250000000002)

We consider the function $g$ defined on the interval $] 0 ; + \infty [$, by
$$g ( x ) = \frac { \ln ( x ) } { 1 + x ^ { 2 } }$$
We admit that $g$ is differentiable on the interval $] 0 ; + \infty \left[ \right.$ and we denote $g ^ { \prime }$ its derivative function. We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in the plane with respect to a coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } )$.
We also consider the function $f$ defined on $]0;+\infty[$ by $f(x) = 1 + x^2 - 2x^2\ln(x)$, and $\alpha$ denotes the unique solution of $f(x)=0$ in $[1;+\infty[$. We admit that $g(\alpha) = \frac{1}{2\alpha^2}$.

1. Prove that for all real $x$ in the interval $] 0 ; + \infty \left[ , \quad g ^ { \prime } ( x ) = \frac { f ( x ) } { x \left( 1 + x ^ { 2 } \right) ^ { 2 } } \right.$.
2. Prove that the function $g$ admits a maximum at $x = \alpha$.
3. We denote $T _ { 1 }$ the tangent line to $\mathscr { C } _ { g }$ at the point with abscissa 1 and we denote $T _ { \alpha }$ the tangent line to $\mathscr { C } _ { g }$ at the point with abscissa $\alpha$. Determine, as a function of $\alpha$, the coordinates of the intersection point of the lines $T _ { 1 }$ and $T _ { \alpha }$.

We consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \ln\left(\mathrm{e}^{2x} - \mathrm{e}^{x} + 1\right).$$ We denote $\mathscr{C}_f$ its representative curve.
A student formulates the following conjectures based on this graphical representation:

1. The equation $f(x) = 2$ seems to admit at least one solution.
2. The largest interval on which the function $f$ seems to be increasing is $[-0{,}5; +\infty[$.
3. The equation of the tangent line at the point with abscissa $x = 0$ seems to be: $y = 1{,}5x$.

Part A: Study of an auxiliary function
We define on $\mathbb{R}$ the function $g$ defined by $$g(x) = \mathrm{e}^{2x} - \mathrm{e}^{x} + 1.$$

1. Determine $\lim_{x \rightarrow -\infty} g(x)$.
2. Show that $\lim_{x \rightarrow +\infty} g(x) = +\infty$.
3. Show that $g'(x) = \mathrm{e}^{x}\left(2\mathrm{e}^{x} - 1\right)$ for all $x \in \mathbb{R}$.
4. Study the monotonicity of the function $g$ on $\mathbb{R}$. Draw up the variation table of the function $g$ showing the exact value of the extrema if any, as well as the limits of $g$ at $-\infty$ and $+\infty$.
5. Deduce the sign of $g$ on $\mathbb{R}$.
6. Without necessarily carrying out the calculations, explain how one could establish the result of question 5 by setting $X = \mathrm{e}^{x}$.

Part B

1. Justify that the function $f$ is well defined on $\mathbb{R}$.
2. The derivative function of the function $f$ is denoted $f'$. Justify that $f'(x) = \frac{g'(x)}{g(x)}$ for all $x \in \mathbb{R}$.
3. Determine an equation of the tangent line to the curve at the point with abscissa 0.
4. Show that the function $f$ is strictly increasing on $[-\ln(2); +\infty[$.
5. Show that the equation $f(x) = 2$ admits a unique solution $\alpha$ on $[-\ln(2); +\infty[$ and determine an approximate value of $\alpha$ to $10^{-2}$ near.

Part C
Using the results of Part B, indicate, for each conjecture of the student, whether it is true or false. Justify.

We consider the function $f$ defined on the interval $]0; +\infty[$ by $$f(x) = (2 - \ln x) \times \ln x,$$ where ln denotes the natural logarithm function.
We admit that the function $f$ is twice differentiable on $]0; +\infty[$.
We denote by $C$ the representative curve of the function $f$ in an orthogonal coordinate system and $C'$ the representative curve of the function $f'$, the derivative function of the function $f$.
The curve $\boldsymbol{C}'$ is given (with its unique horizontal tangent (T)).

1. By graphical reading, with the precision that the above diagram allows, give: a. the slope of the tangent to $C$ at the point with abscissa 1. b. the largest interval on which the function $f$ is convex.
2. a. Calculate the limit of the function $f$ at $+\infty$. b. Calculate $\lim_{x \rightarrow 0} f(x)$. Interpret this result graphically.
3. Show that the curve $C$ intersects the x-axis at exactly two points, whose coordinates you will specify.
4. a. Show that for all real $x$ belonging to $]0; +\infty[$, $f'(x) = \dfrac{2(1 - \ln x)}{x}$. b. Deduce, by justifying, the table of variations of the function $f$ on $]0; +\infty[$.
5. We denote by $f''$ the second derivative of $f$ and we admit that for all real $x$ belonging to $]0; +\infty[$, $f''(x) = \dfrac{2(\ln x - 2)}{x^2}$. Determine by calculation the largest interval on which the function $f$ is convex and specify the coordinates of the inflection point of the curve $C$.

We consider the function $f$ defined on $] 0 ; + \infty [$ by:
$$f ( x ) = 3 x + 1 - 2 x \ln ( x ) .$$
We admit that the function $f$ is twice differentiable on $] 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative. We denote $\mathscr { C } _ { f }$ its representative curve in a coordinate system of the plane.

1. Determine the limit of the function $f$ at 0 and at $+ \infty$.
2. a. Prove that for every strictly positive real number $x$: $f ^ { \prime } ( x ) = 1 - 2 \ln ( x )$. b. Study the sign of $f ^ { \prime }$ and draw up the variation table of the function $f$ on the interval $] 0 ; + \infty [$. This table should include the limits as well as the exact value of the extremum.
3. a. Prove that the equation $f ( x ) = 0$ has a unique solution on $] 0 ; + \infty [$. We denote this solution by $\alpha$. b. Deduce the sign of the function $f$ on $] 0 ; + \infty [$.
4. We consider any primitive of the function $f$ on the interval $] 0$; $+ \infty [$. We denote it by $F$. Can we assert that the function $F$ is strictly decreasing on the interval $\left[ \mathrm { e } ^ { \frac { 1 } { 2 } } ; + \infty [ \right.$ ? Justify.
5. a. Study the convexity of the function $f$ on $] 0 ; + \infty [$. What is the position of the curve $\mathscr { C } _ { f }$ relative to its tangent lines? b. Determine an equation of the tangent line $T$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 1. c. Deduce from questions 5.a and 5.b that for every strictly positive real number $x$: $$\ln ( x ) \geqslant 1 - \frac { 1 } { x } .$$

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, a multiple answer, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.

1. We consider the function $f$ defined on the interval $]1; +\infty[$ by $$f(x) = 0{,}05 - \frac{\ln x}{x-1}$$ The limit of the function $f$ at $+\infty$ is equal to: a. $+\infty$ b. 0.05 c. $-\infty$ d. 0
   
2. We consider a function $h$ continuous on the interval $[-2;4]$ such that: $$h(-1) = 0, \quad h(1) = 4, \quad h(3) = -1.$$ We can affirm that: a. the function $h$ is increasing on the interval $[-1; 1]$. b. the function $h$ is positive on the interval $[-1; 1]$. c. there exists at least one real number $a$ in the interval $[1; 3]$ such that $h(a) = 1$. d. the equation $h(x) = 1$ has exactly two solutions in the interval $[-2; 4]$.
   
3. We consider two sequences $(u_{n})$ and $(v_{n})$ with strictly positive terms such that $\lim_{n \rightarrow +\infty} u_{n} = +\infty$ and $(v_{n})$ converges to 0. We can affirm that: a. the sequence $\left(\dfrac{1}{v_{n}}\right)$ converges. b. the sequence $\left(\dfrac{v_{n}}{u_{n}}\right)$ converges. c. the sequence $(u_{n})$ is increasing. d. $\lim_{n \rightarrow +\infty} \left(-u_{n}\right)^{n} = -\infty$
   
4. To participate in a game, a player must pay $4\,€$. They then roll a fair six-sided die:
   • if they get 1, they win $12\,€$;
   • if they get an even number, they win $3\,€$;
   • otherwise, they win nothing.
   On average, the player: a. wins $3.50\,€$ b. loses $3\,€$. c. loses $1.50\,€$ d. loses $0.50\,€$.
   
5. We consider the random variable $X$ following the binomial distribution $\mathscr{B}(3; p)$. We know that $P(X = 0) = \dfrac{1}{125}$. We can affirm that: a. $p = \dfrac{1}{5}$ b. $P(X = 1) = \dfrac{124}{125}$ c. $p = \dfrac{4}{5}$ d. $P(X = 1) = \dfrac{4}{5}$

Let the function $f$ defined on $\mathbb{R}$ by $$f(x) = \ln\left(1 + \mathrm{e}^{-x}\right) + \frac{1}{4}x.$$ We denote $\mathscr{C}_f$ the representative curve of the function $f$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ of the plane.
Part A

1. Determine the limit of $f$ at $+\infty$.
2. We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. a. Show that, for all real $x$, $f'(x) = \dfrac{\mathrm{e}^x - 3}{4\left(\mathrm{e}^x + 1\right)}$. b. Deduce the variations of the function $f$ on $\mathbb{R}$. c. Show that the equation $f(x) = 1$ admits a unique solution $\alpha$ in the interval $[2;5]$.

Part B
We will admit that the function $f'$ is differentiable on $\mathbb{R}$ and for all real $x$, $$f''(x) = \frac{\mathrm{e}^x}{\left(\mathrm{e}^x + 1\right)^2}.$$ We denote $\Delta$ the tangent line to the curve $\mathscr{C}_f$ at the point with abscissa 0. In the graph below, we have represented the curve $\mathscr{C}_f$, the tangent line $\Delta$, and the quadrilateral MNPQ such that M and N are the two points of the curve $\mathscr{C}_f$ with abscissas $\alpha$ and $-\alpha$ respectively, and Q and P are the two points of the line $\Delta$ with abscissas $\alpha$ and $-\alpha$ respectively.

1. a. Justify the sign of $f''(x)$ for $x \in \mathbb{R}$. b. Deduce that the portion of the curve $\mathscr{C}_f$ on the interval $[-\alpha; \alpha]$ is inscribed in the quadrilateral MNPQ.
2. a. Show that $f(-\alpha) = \ln\left(\mathrm{e}^{-\alpha} + 1\right) + \dfrac{3}{4}\alpha$. b. Prove that the quadrilateral MNPQ is a parallelogram.

We consider the function $f$ defined for every real $x$ in the interval $]0; +\infty[$ by:
$$f(x) = 5x^2 + 2x - 2x^2\ln(x).$$
We denote by $\mathscr{C}_f$ the representative curve of $f$ in an orthogonal reference frame of the plane. We admit that $f$ is twice differentiable on the interval $]0; +\infty[$. We denote by $f'$ its derivative and $f''$ its second derivative.

1. a. Prove that the limit of the function $f$ at 0 is equal to 0. b. Determine the limit of the function $f$ at $+\infty$.
2. Determine $f'(x)$ for every real $x$ in the interval $]0; +\infty[$.
3. a. Prove that for every real $x$ in the interval $]0; +\infty[$ $$f''(x) = 4(1 - \ln(x)).$$ b. Deduce the largest interval on which the curve $\mathscr{C}_f$ is above its tangent lines. c. Draw the variation table of the function $f'$ on the interval $]0; +\infty[$. (We will admit that $\lim_{\substack{x \to 0 \\ x > 0}} f'(x) = 2$ and that $\lim_{x \to +\infty} f'(x) = -\infty$.)
4. a. Show that the equation $f'(x) = 0$ admits in the interval $]0; +\infty[$ a unique solution $\alpha$ for which we will give an enclosure of amplitude $10^{-2}$. b. Deduce the sign of $f'(x)$ on the interval $]0; +\infty[$ as well as the variation table of the function $f$ on the interval $]0; +\infty[$.
5. a. Using the equality $f'(\alpha) = 0$, prove that: $$\ln(\alpha) = \frac{4\alpha + 1}{2\alpha}.$$ Deduce that $f(\alpha) = \alpha^2 + \alpha$. b. Deduce an enclosure of amplitude $10^{-1}$ of the maximum of the function $f$.

Part A: study of the function $\boldsymbol { f }$

The function $f$ is defined on the interval $] 0$; $+ \infty$ [ by:
$$f ( x ) = x - 2 + \frac { 1 } { 2 } \ln x$$
where ln denotes the natural logarithm function. We admit that the function $f$ is twice differentiable on $] 0 ; + \infty \left[ \right.$, we denote by $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative.

1. a. Determine, by justifying, the limits of $f$ at 0 and at $+ \infty$. b. Show that for all $x$ belonging to $] 0$; $+ \infty \left[ \right.$, we have: $f ^ { \prime } ( x ) = \frac { 2 x + 1 } { 2 x }$. c. Study the direction of variation of $f$ on $] 0 ; + \infty [$. d. Study the convexity of $f$ on $] 0 ; + \infty [$.
2. a. Show that the equation $f ( x ) = 0$ admits in $] 0 ; + \infty [$ a unique solution which we denote by $\alpha$ and justify that $\alpha$ belongs to the interval $[ 1 ; 2 ]$. b. Determine the sign of $f ( x )$ for $x \in ] 0$; $+ \infty [$. c. Show that $\ln ( \alpha ) = 2 ( 2 - \alpha )$.

Part B: study of the function $g$

The function $g$ is defined on $] 0 ; 1]$ by:
$$g ( x ) = - \frac { 7 } { 8 } x ^ { 2 } + x - \frac { 1 } { 4 } x ^ { 2 } \ln x .$$
We admit that the function $g$ is differentiable on $] 0 ; 1 ]$ and we denote by $g ^ { \prime }$ its derivative function.

1. Calculate $g ^ { \prime } ( x )$ for $\left. x \in \right] 0$; 1] then verify that $g ^ { \prime } ( x ) = x f \left( \frac { 1 } { x } \right)$.
2. a. Justify that for $x$ belonging to the interval $] 0$; $\frac { 1 } { \alpha } \left[ \right.$, we have $f \left( \frac { 1 } { x } \right) > 0$. b. We admit the following sign table:

$x$	\multicolumn{1}{|c}{0}	$\frac { 1 } { \alpha }$	1
sign of $f \left( \frac { 1 } { x } \right)$		+	0	-

Deduce the variation table of $g$ on the interval $] 0 ; 1 ]$. Images and limits are not required.

Part C: an area calculation

The following are represented on the graph below:

• The curve $\mathscr { C } _ { g }$ of the function $g$;
• The parabola $\mathscr { P }$ with equation $y = - \frac { 7 } { 8 } x ^ { 2 } + x$ on the interval $\left. ] 0 ; 1 \right]$.

We wish to calculate the area $\mathscr { A }$ of the shaded region between the curves $\mathscr { C } _ { g }$ and $\mathscr { P }$, and the lines with equations $x = \frac { 1 } { \alpha }$ and $x = 1$. We recall that $\ln ( \alpha ) = 2 ( 2 - \alpha )$.

1. a. Justify the relative position of the curves $C _ { g }$ and $\mathscr { P }$ on the interval $\left. ] 0 ; 1 \right]$. b. Prove the equality: $$\int _ { \frac { 1 } { \alpha } } ^ { 1 } x ^ { 2 } \ln x \mathrm {~d} x = \frac { - \alpha ^ { 3 } - 6 \alpha + 13 } { 9 \alpha ^ { 3 } }$$
2. Deduce the expression as a function of $\alpha$ of the area $\mathscr { A }$.

Let $f$ be the function defined on $\mathbb { R }$ by
$$f ( x ) = x \mathrm { e } ^ { - 2 x } .$$
We admit that $f$ is twice differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ the derivative of the function $f$. We denote $C _ { f }$ the representative curve of $f$ in an orthonormal coordinate system of the plane.
For each of the following statements, specify whether it is true or false, then justify the answer given.
Any answer without justification will not be taken into account.
Statement 1. For all real $x$, we have $f ^ { \prime } ( x ) = ( - 2 x + 1 ) \mathrm { e } ^ { - 2 x }$.
Statement 2. The function $f$ is a solution on $\mathbb { R }$ of the differential equation:
$$y ^ { \prime } + 2 y = \mathrm { e } ^ { - 2 x }$$
Statement 3. The function $f$ is convex on $] - \infty ; 1 ]$.
Statement 4. The equation $f ( x ) = - 1$ admits a unique solution on $\mathbb { R }$.
Statement 5. The area of the region bounded by the curve $C _ { f }$, the $x$-axis and the lines with equations $x = 0$ and $x = 1$ is equal to $\frac { 1 } { 4 } - \frac { 3 \mathrm { e } ^ { - 2 } } { 4 }$.

We consider the function $f$ defined on the interval $]2; +\infty[$ by $$f(x) = x\ln(x-2)$$ Part of the representative curve $\mathscr{C}_f$ of the function $f$ is given below.

1. Conjecture, using the graph, the direction of variation of $f$, its limits at the boundaries of its domain of definition, and any possible asymptotes.
2. Solve the equation $f(x) = 0$ on $]2; +\infty[$.
3. Calculate $\displaystyle\lim_{\substack{x \rightarrow 2 \\ x > 2}} f(x)$. Does this result confirm one of the conjectures made in question 1?
4. Prove that for all $x$ belonging to $]2; +\infty[$: $$f'(x) = \ln(x-2) + \frac{x}{x-2}$$
5. We consider the function $g$ defined on the interval $]2; +\infty[$ by $g(x) = f'(x)$. a. Prove that for all $x$ belonging to $]2; +\infty[$, we have: $$g'(x) = \frac{x-4}{(x-2)^2}$$ b. We admit that $\displaystyle\lim_{\substack{x \rightarrow 2 \\ x > 2}} g(x) = +\infty$ and that $\displaystyle\lim_{x \rightarrow +\infty} g(x) = +\infty$. Deduce the table of variations of the function $g$ on $]2; +\infty[$. The exact value of the extremum of the function $g$ should be shown. c. Deduce that, for all $x$ belonging to $]2; +\infty[$, $g(x) > 0$. d. Deduce the direction of variation of the function $f$ on $]2; +\infty[$.
6. Study the convexity of the function $f$ on $]2; +\infty[$ and specify the coordinates of any possible inflection point of the representative curve of the function $f$.
7. How many values of $x$ exist for which the representative curve of $f$ admits a tangent with slope equal to 3?

Part A

We consider the function $f$ defined on the interval $]-1; +\infty[$ by $$f(x) = 4\ln(x+1) - \frac{x^2}{25}$$ We admit that the function $f$ is differentiable on the interval $]-1; +\infty[$.

1. Determine the limit of the function $f$ at $-1$.
2. Show that, for all $x$ belonging to the interval $]-1; +\infty[$, we have: $$f'(x) = \frac{100 - 2x - 2x^2}{25(x+1)}$$
3. Study the variations of the function $f$ on the interval $]-1; +\infty[$ and then deduce that the function $f$ is strictly increasing on the interval $[2; 6.5]$.
4. We consider $h$ the function defined on the interval $[2; 6.5]$ by $h(x) = f(x) - x$.
   The table of variations of the function $h$ is given (showing $h$ increases then decreases on $[2;6.5]$ with $h(2) < 0$ and $h(6.5) < 0$ and a positive maximum in between).
   Show that the equation $h(x) = 0$ admits a unique solution $\alpha \in [2; 6.5]$.
5. Consider the following script, written in Python language: \begin{verbatim} from math import * def f(x): return 4*log(1+x)-(x**2)/25 def bornes(n) : p = 1/10**n x = 6 while f(x)-x > 0 : x = x + p return (x-p,x) \end{verbatim} We recall that in Python language:
   • the command $\log(x)$ returns the value $\ln x$;
   • the command $\mathrm{c}**\mathrm{d}$ returns the value of $c^d$.
   a. Give the values returned by the command \texttt{bornes(2)}. The values will be given rounded to the nearest hundredth. b. Interpret these values in the context of the exercise.

Part B

In this part, we may use the results obtained in Part A. We consider the sequence $(u_n)$ defined by $u_0 = 2$, and, for all natural integer $n$, $u_{n+1} = f(u_n)$.

1. Show by induction that for all natural integer $n$, $$2 \leqslant u_n \leqslant u_{n+1} < 6.5.$$
2. Deduce that the sequence $(u_n)$ converges to a limit $\ell$.
3. We recall that the real number $\alpha$, defined in Part A, is the solution of the equation $h(x) = 0$ on the interval $[2; 6.5]$. Justify that $\ell = \alpha$.

We equip the plane with an orthonormal coordinate system. For every natural integer $n$, we consider the function $f_n$ defined on $[0; +\infty[$ by: $$f_0(x) = \mathrm{e}^{-x} \text{ and, for } n \geqslant 1,\ f_n(x) = x^n \mathrm{e}^{-x}.$$ For every natural integer $n$, we denote $\mathscr{C}_n$ the representative curve of the function $f_n$.
Parts A and B are independent.
Part A: Study of the functions $f_n$ for $n \geqslant 1$
We consider a natural integer $n \geqslant 1$.

1. a. We admit that the function $f_n$ is differentiable on $[0; +\infty[$. Show that for all $x \geqslant 0$, $$f_n'(x) = (n - x)x^{n-1}\mathrm{e}^{-x}.$$ b. Justify all elements of the table below:
   

   $x$	0		$n$		$+\infty$
   $f_n'(x)$		+	0	-
   			$\left(\frac{n}{\mathrm{e}}\right)^n$
   $f_n$
   	0				0

   
   
2. Justify by calculation that the point $\mathrm{A}\left(1; \mathrm{e}^{-1}\right)$ belongs to the curve $\mathscr{C}_n$.

Part B: Study of the integrals $\int_0^1 f_n(x)\,\mathrm{d}x$ for $n \geqslant 0$
In this part, we study the functions $f_n$ on $[0; 1]$ and we consider the sequence $(I_n)$ defined for every natural integer $n$ by: $$I_n = \int_0^1 f_n(x)\,\mathrm{d}x = \int_0^1 x^n \mathrm{e}^{-x}\,\mathrm{d}x.$$

1. On the graph in APPENDIX, the curves $\mathscr{C}_0, \mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_{10}$ and $\mathscr{C}_{100}$ are represented. a. Give a graphical interpretation of $I_n$. b. By reading this graph, what conjecture can be made about the limit of the sequence $(I_n)$?
2. Calculate $I_0$.
3. a. Let $n$ be a natural integer. Prove that for all $x \in [0; 1]$, $$0 \leqslant x^{n+1} \leqslant x^n.$$ b. Deduce that for every natural integer $n$, we have: $$0 \leqslant I_{n+1} \leqslant I_n.$$
4. Prove that the sequence $(I_n)$ is convergent, towards a limit greater than or equal to zero that we will denote $\ell$.
5. Using integration by parts, prove that for every natural integer $n$ we have: $$I_{n+1} = (n+1)I_n - \frac{1}{\mathrm{e}}.$$
6. a. Prove that if $\ell > 0$, the equality from question 5 leads to a contradiction. b. Prove that $\ell = 0$. You may use question 6.a.
7. The script of the \texttt{mystere} function is given below, written in Python language. The constant \texttt{e} has been imported. \begin{verbatim} def mystere(n): I = 1 - 1/e L = [I] for i in range(n): I = (i + 1)*I - 1/e L.append(I) return L \end{verbatim} What does \texttt{mystere(100)} return in the context of the exercise?

We denote by $f$ the function defined on the interval $[ 0 ; \pi ]$ by
$$f ( x ) = \mathrm { e } ^ { x } \sin ( x )$$
We denote by $\mathscr { C } _ { f }$ the representative curve of $f$ in a coordinate system.

PART A

1. a. Prove that for every real number $x$ in the interval $[ 0 ; \pi ]$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { x } [ \sin ( x ) + \cos ( x ) ]$$ b. Justify that the function $f$ is strictly increasing on the interval $\left[ 0 ; \frac { \pi } { 2 } \right]$
2. a. Determine an equation of the tangent $T$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0. b. Prove that the function $f$ is convex on the interval $\left[ 0 ; \frac { \pi } { 2 } \right]$. c. Deduce that for every real number $x$ in the interval $\left[ 0 ; \frac { \pi } { 2 } \right] , \mathrm { e } ^ { x } \sin ( x ) \geqslant x$.
3. Justify that the point with abscissa $\frac { \pi } { 2 }$ of the representative curve of the function $f$ is an inflection point.

PART B

We denote
$$I = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { x } \sin ( x ) \mathrm { d } x \text { and } \quad J = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { x } \cos ( x ) \mathrm { d } x$$

1. By integrating by parts the integral $I$ in two different ways, establish the following two relations: $$I = 1 + J \quad \text { and } \quad I = \mathrm { e } ^ { \frac { \pi } { 2 } } - J$$
2. Deduce that $I = \frac { 1 + \mathrm { e } ^ { \frac { \pi } { 2 } } } { 2 }$.
3. We denote by $g$ the function defined on $\mathbb { R }$ by $g ( x ) = x$. Calculate the exact value of the area of the shaded region situated between the curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ and the lines with equations $x = 0$ and $x = \frac { \pi } { 2 }$.

In a laboratory, a chemical reaction is studied in a closed reactor under certain conditions. The numerical processing of experimental data made it possible to model the evolution of the temperature of this chemical reaction as a function of time. Temperature is expressed in degrees Celsius and time is expressed in minutes. Throughout the exercise, we place ourselves on the time interval $[0;10]$.

Part A

In an orthogonal coordinate system of the plane, we give below the representative curve of the temperature function as a function of time on the interval $[0; 10]$.

1. Determine, by graphical reading, after how much time the temperature returns to its initial value at time $t = 0$.

We call $f$ the temperature function represented by the curve above. We specify that the function $f$ is defined and differentiable on the interval $[0; 10]$. We admit that the function $f$ can be written in the form $f(t) = (at + b)\mathrm{e}^{-0.5t}$ where $a$ and $b$ are two real constants.
2. We admit that the exact value of $f(0)$ is 40. Deduce the value of $b$.
3. We admit that $f$ satisfies the differential equation (E): $y' + 0.5y = 60\mathrm{e}^{-0.5t}$. Determine the value of $a$.

Part B: Study of the function $f$

We admit that the function $f$ is defined for every real $t$ in the interval $[0; 10]$ by $$f(t) = (60t + 40)\mathrm{e}^{-0.5t}$$

1. Show that for every real $t$ in the interval $[0; 10]$, we have: $f'(t) = (40 - 30t)\mathrm{e}^{-0.5t}$.
2. a. Study the direction of variation of the function $f$ on the interval $[0; 10]$. Draw the variation table of the function $f$ showing the images of the values present in the table. b. Show that the equation $f(t) = 40$ has a unique solution $\alpha$ strictly positive on the interval $]0; 10]$. c. Give an approximate value of $\alpha$ to the nearest tenth and give an interpretation in the context of the exercise.
3. We define the average temperature, expressed in degrees Celsius, of this chemical reaction between two times $t_{1}$ and $t_{2}$, expressed in minutes, by $$\frac{1}{t_{2} - t_{1}} \int_{t_{1}}^{t_{2}} f(t)\,\mathrm{dt}$$ a. Using integration by parts, show that $$\int_{0}^{4} f(t)\,\mathrm{dt} = 320 - \frac{800}{\mathrm{e}^{2}}$$ b. Deduce an approximate value, to the nearest degree Celsius, of the average temperature of this chemical reaction during the first 4 minutes.

Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by: $$f ( x ) = \frac { \ln ( x ) } { x ^ { 2 } } + 1$$ We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthonormal coordinate system. It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function.

1. Determine the limits of the function $f$ at 0 and at $+ \infty$. Deduce the possible asymptotes to the curve $\mathscr { C } _ { f }$.
2. Show that, for all real $x$ in the interval $] 0 ; + \infty [$, we have: $$f ^ { \prime } ( x ) = \frac { 1 - 2 \ln ( x ) } { x ^ { 3 } }$$
3. Deduce the variation table of the function $f$ on the interval $] 0 ; + \infty [$.
4. a. Show that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $] 0 ; + \infty [$. b. Give an interval for the real number $\alpha$ with amplitude 0.01. c. Deduce the sign of the function $f$ on the interval $] 0 ; + \infty [$.
5. Consider the function $g$ defined on the interval $] 0 ; + \infty [$ by: $$g ( x ) = \ln ( x )$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthonormal coordinate system with origin O. We consider a strictly positive real $x$ and the point M of the curve $\mathscr{C} _ { g }$ with abscissa $x$. We denote OM the distance between points O and M. a. Express the quantity $\mathrm { OM } ^ { 2 }$ as a function of the real $x$. b. Show that, when the real $x$ ranges over the interval $] 0 ; + \infty [$, the quantity $\mathrm { OM } ^ { 2 }$ admits a minimum at $\alpha$. c. The minimum value of the distance OM, when the real $x$ ranges over the interval $] 0 ; + \infty [$, is called the distance from point O to the curve $\mathscr { C } _ { g }$. We denote $d$ this distance. Express $d$ in terms of $\alpha$.

We consider $f$ the function defined on the interval $] 0 ; + \infty \left[ \right.$ by $f ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } } { 2 \sqrt { x } }$ and we call $C _ { f }$ its representative curve in an orthonormal coordinate system.

1. We define the function $g$ on the interval $] 0 ; + \infty \left[ \operatorname { by } g ( x ) = \mathrm { e } ^ { \sqrt { x } } \right.$.
   (a) Show that $g ^ { \prime } ( x ) = f ( x )$ for all $x$ in the interval $] 0 ; + \infty [$.
   (b) For all real $x$ in the interval $] 0 ; + \infty \left[ \right.$, calculate $f ^ { \prime } ( x )$ and show that:

$$f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( \sqrt { x } - 1 ) } { 4 x \sqrt { x } }$$

1. (a) Determine the limit of the function $f$ at 0.
   (b) Interpret this result graphically.
2. (a) Determine the limit of the function $f$ at $+ \infty$.
   (b) Study the direction of variation of the function $f$ on $] 0 ; + \infty [$. Draw the variation table of the function $f$ including the limits at the boundaries of the domain of definition.
   (c) Prove that the equation $f ( x ) = 2$ has a unique solution on the interval [ $1 ; + \infty [$ and give an approximate value to $10 ^ { - 1 }$ of this solution.
3. We set $I = \int _ { 1 } ^ { 2 } f ( x ) d x$.
   (a) Calculate $I$.
   (b) Interpret graphically the result obtained.
4. We admit that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$ and that:
   $$f ^ { \prime \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( x - 3 \sqrt { x } + 3 ) } { 8 x ^ { 2 } \sqrt { x } }$$
   (a) By setting $X = \sqrt { x }$, show that $x - 3 \sqrt { x } + 3 > 0$ for all real $x$ in the interval $] 0 ; + \infty [$.
   (b) Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$.