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 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 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 $f$.

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 $] 0 ; + \infty [$.

Consider the function $f$ defined on $\mathbb { R }$ by :
$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - 3 x } }$$
We denote $\mathscr { C } _ { f }$ its representative curve in an orthogonal coordinate system of the plane. We name A the point with coordinates $\left( 0 ; \frac { 1 } { 2 } \right)$ and B the point with coordinates $\left( 1 ; \frac { 5 } { 4 } \right)$. Below we have drawn the curve $\mathscr { C } _ { f }$ and $\mathscr { T }$ the tangent line to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0.

Part A: graphical readings

In this part, results will be obtained by graphical reading. No justification is required.

1. Determine the reduced equation of the tangent line $\mathscr { T }$.
2. Give the intervals on which the function $f$ appears to be convex or concave.

Part B : study of the function

1. We admit that the function $f$ is differentiable on $\mathbb { R }$.

Determine the expression of its derivative function $f ^ { \prime }$.
2. Justify that the function $f$ is strictly increasing on $\mathbb { R }$.
3. a. Determine the limit at $+ \infty$ of the function $f$. b. Determine the limit at $- \infty$ of the function $f$.
4. Determine the exact value of the solution $\alpha$ of the equation $f ( x ) = 0,99$.

Part C : Tangent line and convexity

1. Determine by calculation an equation of the tangent line $\mathscr { T }$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0.

We admit that the function $f$ is twice differentiable on $\mathbb { R }$. We denote $f ^ { \prime \prime }$ the second derivative function of the function $f$. We admit that $f ^ { \prime \prime }$ is defined on $\mathbb { R }$ by:
$$f ^ { \prime \prime } ( x ) = \frac { 9 \mathrm { e } ^ { - 3 x } \left( \mathrm { e } ^ { - 3 x } - 1 \right) } { \left( 1 + \mathrm { e } ^ { - 3 x } \right) ^ { 3 } } .$$

1. Study the sign of the function $f ^ { \prime \prime }$ on $\mathbb { R }$.
2. a. Indicate, by justifying, on which interval(s) the function $f$ is convex. b. What does point A represent for the curve $\mathscr { C } _ { f }$ ? c. Deduce the relative position of the tangent line $\mathscr { T }$ and the curve $\mathscr { C } _ { f }$. Justify the answer.

We consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \ln\left(1 + \mathrm{e}^{-x}\right),$$ where $\ln$ denotes the natural logarithm function. We denote by $\mathscr{C}$ its representative curve in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

1. a. Determine the limit of the function $f$ at $-\infty$. b. Determine the limit of the function $f$ at $+\infty$. Interpret this result graphically. c. We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative function. Calculate $f'(x)$ then show that, for every real number $x$, $f'(x) = \frac{-1}{1 + \mathrm{e}^x}$. d. Draw the complete table of variations of the function $f$ on $\mathbb{R}$.
2. We denote by $T_0$ the tangent line to the curve $\mathscr{C}$ at its point with abscissa 0. a. Determine an equation of the tangent line $T_0$. b. Show that the function $f$ is convex on $\mathbb{R}$. c. Deduce that, for every real number $x$, we have: $$f(x) \geqslant -\frac{1}{2}x + \ln(2)$$
3. For every real number $a$ different from 0, we denote by $M_a$ and $N_a$ the points of the curve $\mathscr{C}$ with abscissas $-a$ and $a$ respectively. We therefore have: $M_a(-a; f(-a))$ and $N_a(a; f(a))$. a. Show that, for every real number $x$, we have: $f(x) - f(-x) = -x$. b. Deduce that the lines $T_0$ and $(M_a N_a)$ are parallel.

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. No justification is required. A wrong answer, no answer, or multiple answers, neither gives nor takes away points.

1. We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = 2x\mathrm{e}^x$.
   The number of solutions on $\mathbb{R}$ of the equation $f(x) = -\dfrac{73}{100}$ is equal to:
   a. 0
   b. 1
   c. 2
   d. infinitely many.
   
2. We consider the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \frac{x+1}{\mathrm{e}^x}.$$ The limit of the function $g$ at $-\infty$ is equal to:
   a. $-\infty$
   b. $+\infty$
   c. $0$
   d. it does not exist.
   
3. We consider the function $h$ defined on $\mathbb{R}$ by: $$h(x) = (4x - 16)\mathrm{e}^{2x}.$$ We denote $\mathscr{C}_h$ the representative curve of $h$ in an orthogonal coordinate system. We can affirm that:
   a. $h$ is convex on $\mathbb{R}$.
   b. $\mathscr{C}_h$ has an inflection point at $x = 3$.
   c. $h$ is concave on $\mathbb{R}$.
   d. $\mathscr{C}_h$ has an inflection point at $x = 3.5$.
   
4. We consider the function $k$ defined on the interval $]0; +\infty[$ by: $$k(x) = 3\ln(x) - x.$$ We denote $\mathscr{C}$ the representative curve of the function $k$ in an orthonormal coordinate system. We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point with abscissa $x = \mathrm{e}$. An equation of $T$ is:
   a. $y = (3 - \mathrm{e})x$
   b. $y = \left(\dfrac{3 - \mathrm{e}}{\mathrm{e}}\right)x$
   c. $y = \left(\dfrac{3}{\mathrm{e}} - 1\right)x + 1$
   d. $y = (\mathrm{e} - 1)x + 1$
   
5. We consider the equation $[\ln(x)]^2 + 10\ln(x) + 21 = 0$, with $x \in ]0; +\infty[$.
   The number of solutions of this equation is equal to:
   a. 0
   b. 1
   c. 2
   d. infinitely many.

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 $\ell$.

We consider the function $f$ defined on the interval $]-\infty; 1[$ by $$f(x) = \frac{\mathrm{e}^x}{x-1}$$ We admit that the function $f$ is differentiable on the interval $]-\infty; 1[$. We call $\mathscr{C}$ its representative curve in a coordinate system.

1. 1. [a.] Determine the limit of the function $f$ at 1.
   2. [b.] Deduce from this a graphical interpretation.
2. Determine the limit of the function $f$ at $-\infty$.
3. 1. [a.] Show that for every real number $x$ in the interval $]-\infty; 1[$, we have $$f'(x) = \frac{(x-2)\mathrm{e}^x}{(x-1)^2}$$
   2. [b.] Draw up, by justifying, the table of variations of the function $f$ on the interval $]-\infty; 1[$.
4. We admit that for every real number $x$ in the interval $]-\infty; 1[$, we have $$f''(x) = \frac{\left(x^2 - 4x + 5\right)\mathrm{e}^x}{(x-1)^3}.$$
   1. [a.] Study the convexity of the function $f$ on the interval $]-\infty; 1[$.
   2. [b.] Determine the reduced equation of the tangent line $T$ to the curve $\mathscr{C}$ at the point with abscissa 0.
   3. [c.] Deduce from this that, for every real number $x$ in the interval $]-\infty; 1[$, we have: $$\mathrm{e}^x \geqslant (-2x-1)(x-1).$$
5. 1. [a.] Justify that the equation $f(x) = -2$ admits a unique solution $\alpha$ on the interval $]-\infty; 1[$.
   2. [b.] Using a calculator, determine an interval containing $\alpha$ with amplitude $10^{-2}$.

(Calculus) For the function $f(x) = 4\ln x + \ln(10 - x)$, which of the following statements in are correct? [3 points]
ㄱ. The maximum value of function $f(x)$ is $13\ln 2$. ㄴ. The equation $f(x) = 0$ has two distinct real roots. ㄷ. The graph of function $y = e^{f(x)}$ is concave downward on the interval $(4, 8)$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

If the total maximum value of the function $f ( x ) = \left( \frac { \sqrt { 3 e } } { 2 \sin x } \right) ^ { \sin ^ { 2 } x } , x \in \left( 0 , \frac { \pi } { 2 } \right)$, is $\frac { k } { e }$, then $\left( \frac { k } { e } \right) ^ { 8 } + \frac { k ^ { 8 } } { e ^ { 5 } } + k ^ { 8 }$ is equal to
(1) $e ^ { 3 } + e ^ { 6 } + e ^ { 11 }$
(2) $e ^ { 5 } + e ^ { 6 } + e ^ { 11 }$
(3) $e ^ { 3 } + e ^ { 6 } + e ^ { 10 }$
(4) $e ^ { 3 } + e ^ { 5 } + e ^ { 11 }$