Throughout what follows, $m$ denotes any real number.
Part A
Let $f$ be the function defined and differentiable on the set of real numbers $\mathbb{R}$ such that: $$f(x) = (x+1)\mathrm{e}^x$$

1. Calculate the limit of $f$ at $+\infty$ and $-\infty$.
2. We denote by $f'$ the derivative function of $f$ on $\mathbb{R}$. Prove that for all real $x$, $f'(x) = (x+2)\mathrm{e}^x$.
3. Draw the variation table of $f$ on $\mathbb{R}$.

Part B
We define the function $g_m$ on $\mathbb{R}$ by: $$g_m(x) = x + 1 - m\mathrm{e}^{-x}$$ and we denote by $\mathscr{C}_m$ the curve of function $g_m$ in a frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$ of the plane.

1. a. Prove that $g_m(x) = 0$ if and only if $f(x) = m$. b. Deduce from Part $A$, without justification, the number of intersection points of curve $\mathscr{C}_m$ with the $x$-axis as a function of the real number $m$.
2. We have represented in appendix 2 the curves $\mathscr{C}_0$, $\mathscr{C}_{\mathrm{e}}$, and $\mathscr{C}_{-\mathrm{e}}$ (obtained by taking respectively for $m$ the values 0, e and $-$e). Identify each of these curves in the figure of appendix 2 by justifying.
3. Study the position of curve $\mathscr{C}_m$ relative to the line $\mathscr{D}$ with equation $y = x + 1$ according to the values of the real number $m$.
4. a. We call $D_2$ the part of the plane between curves $\mathscr{C}_{\mathrm{e}}$, $\mathscr{C}_{-\mathrm{e}}$, the axis $(Oy)$ and the line $x = 2$. Shade $D_2$ on appendix 2. b. In this question, $a$ denotes a positive real number, $D_a$ the part of the plane between $\mathscr{C}_{\mathrm{e}}$, $\mathscr{C}_{-\mathrm{e}}$, the axis $(Oy)$ and the line $\Delta_a$ with equation $x = a$. We denote by $\mathscr{A}(a)$ the area of this part of the plane, expressed in square units. Prove that for all positive real $a$: $\mathscr{A}(a) = 2\mathrm{e} - 2\mathrm{e}^{1-a}$. Deduce the limit of $\mathscr{A}(a)$ as $a$ tends to $+\infty$.

For each of the following propositions, indicate whether it is true or false and justify your chosen answer. One point is awarded for each correct answer that is properly justified. An answer without justification is not taken into account. An absence of an answer is not penalized.
Proposition 1 Every positive increasing sequence tends to $+ \infty$.
Proposition 2 $g$ is the function defined on $] - \frac { 1 } { 2 } ; + \infty [$ by $$g ( x ) = 2 x \ln ( 2 x + 1 ) .$$ On $] - \frac { 1 } { 2 } ; + \infty$ [, the equation $g ( x ) = 2 x$ has a unique solution: $\frac { \mathrm { e } - 1 } { 2 }$.
Proposition 3 The slope of the tangent line to the curve representing the function $g$ at the point with abscissa $\frac { 1 } { 2 }$ is: $1 + \ln 4$.
Proposition 4 Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ). $\mathscr { P }$ and $\mathscr { R }$ are the planes with equations respectively: $2 x + 3 y - z - 11 = 0$ and $x + y + 5 z - 11 = 0$. The planes $\mathscr { P }$ and $\mathscr { R }$ intersect perpendicularly.

In this exercise, the plane is equipped with an orthonormal coordinate system.
The curve with equation is represented below: $$y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 2 \right) .$$ This curve is called a ``catenary''.
We are interested here in ``catenary arcs'' delimited by two points of this curve that are symmetric with respect to the $y$-axis. Such an arc is represented on the graph below in solid line. We define the ``width'' and ``height'' of the catenary arc delimited by the points $M$ and $M^{\prime}$ as indicated on the graph.
The purpose of the exercise is to study the possible positions on the curve of the point $M$ with strictly positive abscissa so that the width of the catenary arc is equal to its height.

1. Justify that the problem studied reduces to finding the strictly positive solutions of the equation $$( E ) : \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 4 x - 2 = 0$$
   
2. Let $f$ be the function defined on the interval $[ 0 ; + \infty [$ by: $$f ( x ) = \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 4 x - 2 .$$ a. Verify that for all $x > 0 , f ( x ) = x \left( \frac { \mathrm { e } ^ { x } } { x } - 4 \right) + \mathrm { e } ^ { - x } - 2$. b. Determine $\lim _ { x \rightarrow + \infty } f ( x )$.
   
3. a. Let $f ^ { \prime }$ denote the derivative function of $f$. Calculate $f ^ { \prime } ( x )$, where $x$ belongs to the interval $[ 0 ; + \infty [$. b. Show that the equation $f ^ { \prime } ( x ) = 0$ is equivalent to the equation: $\left( \mathrm { e } ^ { x } \right) ^ { 2 } - 4 \mathrm { e } ^ { x } - 1 = 0$. c. By setting $X = \mathrm { e } ^ { x }$, show that the equation $f ^ { \prime } ( x ) = 0$ has as its unique real solution the number $\ln ( 2 + \sqrt { 5 } )$.
   
4. The sign table of the derivative function $f ^ { \prime }$ of $f$ is given below:
   

   $x$	0		$\ln ( 2 + \sqrt { 5 } )$	$+ \infty$
   $f ^ { \prime } ( x )$	-	0	+

   
   a. Draw up the variation table of the function $f$. b. Prove that the equation $f ( x ) = 0$ has a unique strictly positive solution which we denote by $\alpha$.
   
5. Consider the following algorithm where the variables $a$, $b$ and $m$ are real numbers: \begin{verbatim} While $b - a > 0.1$ do: $m \leftarrow \frac { a + b } { 2 }$ If $\mathrm { e } ^ { m } + \mathrm { e } ^ { - m } - 4 m - 2 > 0$, then: $b \leftarrow m$ Else: $a \leftarrow m$ End If End While \end{verbatim} a. Before execution of this algorithm, the variables $a$ and $b$ contain respectively the values 2 and 3. What do they contain at the end of the algorithm execution? Justify the answer by reproducing and completing the table opposite with the different values taken by the variables at each step of the algorithm.
   

   $m$	$a$	$b$	$b - a$
   	2	3	1
   2.5
   $\ldots$	$\ldots$	$\ldots$

   
   b. How can we use the values obtained at the end of the algorithm in the previous question?
   
6. The width of the Gateway Arch arc, expressed in metres, is equal to twice the strictly positive solution of the equation: $$\left( E ^ { \prime } \right) : \mathrm { e } ^ { \frac { t } { 39 } } + \mathrm { e } ^ { - \frac { t } { 39 } } - 4 \frac { t } { 39 } - 2 = 0$$ Give a bound for the height of the Gateway Arch.

Exercise 4 — Theme: Functions, Exponential Function, Logarithm Function; Sequences
Part A Consider the function $f$ defined for every real $x$ in $]0; 1]$ by: $$f(x) = \mathrm{e}^{-x} + \ln(x).$$

1. Calculate the limit of $f$ at 0.
2. It is admitted that $f$ is differentiable on $]0; 1]$. Let $f'$ denote its derivative function. Prove that, for every real $x$ belonging to $]0; 1]$, we have: $$f'(x) = \frac{1 - x\mathrm{e}^{-x}}{x}$$
3. Justify that, for every real $x$ belonging to $]0; 1]$, we have $x\mathrm{e}^{-x} < 1$. Deduce the variation table of $f$ on $]0; 1]$.
4. Prove that there exists a unique real $\ell$ belonging to $]0; 1]$ such that $f(\ell) = 0$.

Part B

1. Two sequences $(a_n)$ and $(b_n)$ are defined by: $$\left\{\begin{array}{l} a_0 = \frac{1}{10} \\ b_0 = 1 \end{array}\right. \text{ and, for every natural number } n, \left\{\begin{array}{l} a_{n+1} = \mathrm{e}^{-b_n} \\ b_{n+1} = \mathrm{e}^{-a_n} \end{array}\right.$$ a. Calculate $a_1$ and $b_1$. Approximate values to $10^{-2}$ will be given. b. Consider below the function terms, written in Python language. \begin{verbatim} def termes (n) : a=1/10 b=1 for k in range(0,n) : c= ... b = ... a = c return(a,b) \end{verbatim} Copy and complete without justification the box above so that the function termes calculates the terms of the sequences $(a_n)$ and $(b_n)$.
2. Recall that the function $x \longmapsto \mathrm{e}^{-x}$ is decreasing on $\mathbb{R}$. a. Prove by induction that, for every natural number $n$, we have: $$0 < a_n \leqslant a_{n+1} \leqslant b_{n+1} \leqslant b_n \leqslant 1$$ b. Deduce that the sequences $(a_n)$ and $(b_n)$ are convergent.
3. Let $A$ denote the limit of $(a_n)$ and $B$ denote the limit of $(b_n)$. It is admitted that $A$ and $B$ belong to the interval $]0; 1]$, and that $A = \mathrm{e}^{-B}$ and $B = \mathrm{e}^{-A}$. a. Prove that $f(A) = 0$. b. Determine $A - B$.

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 $\ln ( x ) = k x$.

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 $]0;+\infty[$.

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.

Given the function $f ( x ) = \mathrm { e } ^ { x } - a ( x + 2 )$ .
(1) When $a = 1$ , discuss the monotonicity of $f ( x )$ ;
(2) If $f ( x )$ has two zeros, find the range of values for $a$ .

Given the function $f ( x ) = x ^ { 3 } - k x + k ^ { 2 }$ .
(1) Discuss the monotonicity of $f ( x )$;
(2) If $f ( x )$ has three zeros, find the range of values for $k$ .

Given the function $f ( x ) = \frac { e ^ { x } } { x } - \ln x + x - a$.
(1) If $f ( x ) \geq 0$, find the range of values for $a$;
(2) Prove that if $f ( x )$ has two zeros $x _ { 1 }$ and $x _ { 2 }$, then $x _ { 1 } x _ { 2 } < 1$.

Given the function $f ( x ) = a x - \frac { 1 } { x } - ( a + 1 ) \ln x$ .
(1) When $a = 0$ , find the maximum value of $f ( x )$ ;
(2) If $f ( x )$ has exactly one zero point, find the range of values for $a$ .

Given the function $f(x) = \ln(1+x) - x + \frac{1}{2}x^2 - kx^3$, where $0 < k < \frac{1}{3}$.
(1) Prove: $f(x)$ has a unique extremum point and a unique zero point on the interval $(0, +\infty)$;
(2) Let $x_1, x_2$ be the extremum point and zero point of $f(x)$ on the interval $(0, +\infty)$ respectively.
(i) Let $g(t) = f(x_1 + t) - f(x_1 - t)$. Prove: $g(t)$ is monotonically decreasing on the interval $(0, x_1)$;
(ii) Compare the sizes of $2x_1$ and $x_2$, and prove your conclusion.

Let $f ( x ) = x \sin \pi x , x > 0$. Then for all natural numbers $n , f ^ { \prime } ( x )$ vanishes at
(A) a unique point in the interval $\left( n , n + \frac { 1 } { 2 } \right)$
(B) a unique point in the interval $\left( n + \frac { 1 } { 2 } , n + 1 \right)$
(C) a unique point in the interval $( n , n + 1 )$
(D) two points in the interval $( n , n + 1 )$

Consider a quadratic equation $a x ^ { 2 } + b x + c = 0$, where $2 a + 3 b + 6 c = 0$ and let $g ( x ) = a \frac { x ^ { 3 } } { 3 } + b \frac { x ^ { 2 } } { 2 } + c x$. Statement 1: The quadratic equation has at least one root in the interval $( 0,1 )$. Statement 2: The Rolle's theorem is applicable to function $g ( x )$ on the interval $[ 0,1 ]$.
(1) Statement 1 is false, Statement 2 is true.
(2) Statement 1 is true, Statement 2 is false.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.