EXERCISE B Main topics covered: Natural logarithm function, differentiation
This exercise consists of two parts. Some results from the first part will be used in the second.

Part 1: Study of an auxiliary function

Let the function $f$ defined on the interval $[1 ; 4]$ by: $$f ( x ) = - 30 x + 50 + 35 \ln x$$

1. Recall that $f ^ { \prime }$ denotes the derivative function of the function $f$. a. For every real number $x$ in the interval $[ 1 ; 4 ]$, show that: $$f ^ { \prime } ( x ) = \frac { 35 - 30 x } { x }$$ b. Draw up the sign table of $f ^ { \prime } ( x )$ on the interval $[ 1 ; 4 ]$. c. Deduce the variations of $f$ on this same interval.
2. Justify that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $[1;4]$ then give an approximate value of $\alpha$ to $10 ^ { - 3 }$ near.
3. Draw up the sign table of $f ( x )$ for $x \in [ 1 ; 4 ]$.

Part 2: Optimisation

A company sells fruit juice. For $x$ thousand litres sold, with $x$ a real number in the interval $[ 1 ; 4 ]$, the analysis of sales leads to modelling the profit $B ( x )$ by the expression given in thousands of euros by: $$B ( x ) = - 15 x ^ { 2 } + 15 x + 35 x \ln x$$

1. According to the model, calculate the profit made by the company when it sells 2500 litres of fruit juice. Give an approximate value to the nearest euro of this profit.
2. For all $x$ in the interval $[ 1 ; 4 ]$, show that $B ^ { \prime } ( x ) = f ( x )$ where $B ^ { \prime }$ denotes the derivative function of $B$.
3. a. Using the results from part 1, give the variations of the function $B$ on the interval $[1;4]$. b. Deduce the quantity of fruit juice, to the nearest litre, that the company must sell in order to achieve maximum profit.

Consider the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{-2x}$$ Let $f^{\prime\prime}$ denote the second derivative of the function $f$. For any real number $x$, $f^{\prime\prime}(x)$ is equal to: a. $(1-2x)\mathrm{e}^{-2x}$ b. $4(x-1)\mathrm{e}^{-2x}$ c. $4\mathrm{e}^{-2x}$ d. $(x+2)\mathrm{e}^{-2x}$

Let $f$ be the function defined for all real numbers $x$ in the interval $] 0 ; + \infty [$ by:
$$f ( x ) = \frac { \mathrm { e } ^ { 2 x } } { x }$$
The expression of the second derivative $f ^ { \prime \prime }$ of $f$ is given, defined on the interval $] 0 ; + \infty [$ by:
$$f ^ { \prime \prime } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } - 2 x + 1 \right) } { x ^ { 3 } } .$$

1. The function $f ^ { \prime }$, the derivative of $f$, is defined on the interval $] 0 ; + \infty [$ by: a. $f ^ { \prime } ( x ) = 2 \mathrm { e } ^ { 2 x }$ b. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( x - 1 ) } { x ^ { 2 } }$ c. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 2 x - 1 ) } { x ^ { 2 } }$ d. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 1 + 2 x ) } { x ^ { 2 } }$.
2. The function $f$: a. is decreasing on $] 0 ; + \infty [$ b. is monotonic on $] 0 ; + \infty [$ c. admits a minimum at $\frac { 1 } { 2 }$ d. admits a maximum at $\frac { 1 } { 2 }$.
3. The function $f$ has the following limit as $x \to + \infty$: a. $+ \infty$ b. 0 c. 1 d. $\mathrm { e } ^ { 2 x }$.
4. The function $f$: a. is concave on $] 0 ; + \infty [$ b. is convex on $] 0 ; + \infty [$ c. is concave on $] 0 ; \frac { 1 } { 2 } ]$ d. is represented by a curve admitting an inflection point.

The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We admit that the function $f$ represented above is defined on $\mathbb { R }$ by $f ( x ) = ( a x + b ) \mathrm { e } ^ { x }$, where $a$ and $b$ are two real numbers and that its curve intersects the x-axis at the point with coordinates ($-0.5$; 0). We can assert that: a. $a = 10$ and $b = 5$ b. $a = 2.5$ and $b = -0.5$ c. $a = -1.5$ and $b = 5$ d. $a = 0$ and $b = 5$

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 (7 points) Themes: exponential function, sequences In the context of a clinical trial, two treatment protocols for a disease are being considered. The objective of this exercise is to study, for these two protocols, the evolution of the quantity of medication present in a patient's blood as a function of time.
Parts $A$ and $B$ are independent

Part A: Study of the first protocol

The first protocol consists of having the patient take a medication in tablet form. The quantity of medication present in the patient's blood, expressed in mg, is modelled by the function $f$ defined on the interval $[0; 10]$ by $$f(t) = 3t \mathrm{e}^{-0.5t + 1},$$ where $t$ denotes the time, expressed in hours, elapsed since taking the tablet.

1. a. It is admitted that the function $f$ is differentiable on the interval $[0; 10]$ and we denote $f'$ its derivative function. Show that, for every real number $t$ in $[0; 10]$, we have: $f'(t) = 3(-0.5t + 1)\mathrm{e}^{-0.5t + 1}$. b. Deduce the table of variations of the function $f$ on the interval $[0; 10]$. c. According to this model, after how much time will the quantity of medication present in the patient's blood be maximum? What is this maximum quantity?
2. a. Show that the equation $f(t) = 5$ admits a unique solution on the interval $[0; 2]$ denoted $\alpha$, of which you will give an approximate value to $10^{-2}$ near. It is admitted that the equation $f(t) = 5$ admits a unique solution on the interval $[2; 10]$, denoted $\beta$, and that an approximate value of $\beta$ to $10^{-2}$ near is 3.46. b. It is considered that this treatment is effective when the quantity of medication present in the patient's blood is greater than or equal to 5 mg. Determine, to the nearest minute, the duration of effectiveness of the medication in the case of this protocol.

Part B: Study of the second protocol

The second protocol consists of initially injecting the patient, by intravenous injection, a dose of 2 mg of medication and then re-injecting every hour a dose of $1.8$ mg. It is assumed that the medication diffuses instantaneously into the blood and is then progressively eliminated. It is estimated that when one hour has elapsed after an injection, the quantity of medication in the blood has decreased by $30\%$ compared to the quantity present immediately after this injection. This situation is modelled using the sequence $(u_n)$ where, for every natural number $n$, $u_n$ denotes the quantity of medication, expressed in mg, present in the patient's blood immediately after the injection at the $n$-th hour. We therefore have $u_0 = 2$.

1. Calculate, according to this model, the quantity $u_1$, of medication (in mg) present in the patient's blood immediately after the injection at the first hour.
2. Justify that, for every natural number $n$, we have: $u_{n+1} = 0.7u_n + 1.8$.
3. a. Show by induction that, for every natural number $n$, we have: $u_n \leqslant u_{n+1} < 6$. b. Deduce that the sequence $(u_n)$ is convergent. We denote its limit by $\ell$. c. Determine the value of $\ell$. Interpret this value in the context of the exercise.
4. Consider the sequence $(v_n)$ defined, for every natural number $n$, by $v_n = 6 - u_n$. a. Show that the sequence $(v_n)$ is a geometric sequence with ratio 0.7 and specify its first term. b. Determine the expression of $v_n$ as a function of $n$, then of $u_n$ as a function of $n$. c. With this protocol, injections are stopped when the quantity of medication present in the patient's blood is greater than or equal to $5.5$ mg. Determine, by detailing the calculations, the number of injections carried out when applying this protocol.

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.

Exercise 2: Functions, logarithm function
The purpose of this exercise is to study the function $f$, defined on $]0; +\infty[$, by: $$f(x) = 3x - x\ln(x) - 2\ln(x).$$
PART A: Study of an auxiliary function $g$
Let $g$ be the function defined on $]0; +\infty[$ by $$g(x) = 2(x-1) - x\ln(x)$$ We denote $g'$ the derivative function of $g$. We admit that $\lim_{x \rightarrow +\infty} g(x) = -\infty$.

1. Calculate $g(1)$ and $g(\mathrm{e})$.
2. Determine $\lim_{x \rightarrow 0^+} g(x)$ by justifying your approach.
3. Show that, for all $x > 0$, $g'(x) = 1 - \ln(x)$. Deduce the variation table of $g$ on $]0; +\infty[$.
4. Show that the equation $g(x) = 0$ has exactly two distinct solutions on $]0; +\infty[$: 1 and $\alpha$ with $\alpha$ belonging to the interval $[\mathrm{e}; +\infty[$. Give an approximation of $\alpha$ to 0.01.
5. Deduce the sign table of $g$ on $]0; +\infty[$.

PART B: Study of the function $f$
We consider in this part the function $f$, defined on $]0; +\infty[$, by $$f(x) = 3x - x\ln(x) - 2\ln(x).$$ We denote $f'$ the derivative function of $f$. We admit that: $\lim_{x \rightarrow 0^+} f(x) = +\infty$.

1. Determine the limit of $f$ at $+\infty$ by justifying your approach.
2. a. Justify that for all $x > 0$, $f'(x) = \dfrac{g(x)}{x}$. b. Deduce the variation table of $f$ on $]0; +\infty[$.
3. We admit that, for all $x > 0$, the second derivative of $f$, denoted $f''$, is defined by $f''(x) = \dfrac{2-x}{x^2}$. Study the convexity of $f$ and specify the coordinates of the inflection point of $\mathscr{C}_f$.

Part A

In the orthonormal coordinate system above, the representative curves of a function $f$ and its derivative function, denoted $f ^ { \prime }$, are drawn, both defined on $] 3 ; + \infty [$.

1. Associate each curve with the function it represents. Justify.
2. Determine graphically the possible solution(s) of the equation $f ( x ) = 3$.
3. Indicate, by graphical reading, the convexity of the function $f$.

Part B

1. Justify that the quantity $\ln \left( x ^ { 2 } - x - 6 \right)$ is well defined for values $x$ in the interval ]3; $+ \infty$ [, which we will call $I$ in the following.
2. We admit that the function $f$ from Part A is defined by $f ( x ) = \ln \left( x ^ { 2 } - x - 6 \right)$ on $I$. Calculate the limits of the function $f$ at the two endpoints of the interval $I$. Deduce an equation of an asymptote to the representative curve of the function $f$ on $I$.
3. a. Calculate $f ^ { \prime } ( x )$ for all $x$ belonging to $I$. b. Study the direction of variation of the function $f$ on $I$.

Draw the variation table of the function $f$ showing the limits at the endpoints of $I$.
4. a. Justify that the equation $f ( x ) = 3$ admits a unique solution $\alpha$ on the interval ]5; 6[. b. Determine, using a calculator, an approximation of $\alpha$ to within $10 ^ { - 2 }$.
5. a. Justify that $f ^ { \prime \prime } ( x ) = \frac { - 2 x ^ { 2 } + 2 x - 13 } { \left( x ^ { 2 } - x - 6 \right) ^ { 2 } }$. b. Study the convexity of the function $f$ on $I$.

Consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = x ^ { 3 } \mathrm { e } ^ { x }$$
It is admitted that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function.

1. The sequence $(u _ { n })$ is defined by $u _ { 0 } = - 1$ and, for all natural integer $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$. a. Calculate $u _ { 1 }$ then $u _ { 2 }$.
   Exact values will be given, then approximate values to $10 ^ { - 3 }$. b. Consider the function fonc, written in Python language below.
   Recall that in Python language, ``i in range (n)'' means that i varies from 0 to n -1.
   \begin{verbatim} def fonc (n): u =- 1 for i in range(n) : u=u**3*exp(u) return u \end{verbatim}
   Determine, without justification, the value returned by fonc (2) rounded to $10 ^ { - 3 }$.
2. a. Prove that, for all real $x$, we have $f ^ { \prime } ( x ) = x ^ { 2 } \mathrm { e } ^ { x } ( x + 3 )$. b. Justify that the variation table of $f$ on $\mathbb { R }$ is the one represented below:
   

   $x$	$- \infty$	- 3	$+ \infty$
   	0		$+ \infty$
   $f$			$+ 27 \mathrm { e } ^ { - 3 }$

   
   c. Prove, by induction, that for all natural integer $n$, we have:
   $$- 1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 0$$
   d. Deduce that the sequence $(u _ { n })$ is convergent. e. We denote $\ell$ the limit of the sequence $(u _ { n })$.
   Recall that $\ell$ is a solution of the equation $f ( x ) = x$. Determine $\ell$. (For this, it will be admitted that the equation $x ^ { 2 } \mathrm { e } ^ { x } - 1 = 0$ has only one solution in $\mathbb { R }$ and that this solution is strictly greater than $\frac { 1 } { 2 }$).

Exercise 2 — Main topics covered: functions, logarithm function.
Let $f$ be the function defined on the interval $]0; +\infty[$ by: $$f(x) = x\ln(x) - x - 2.$$ We admit that the function $f$ is twice differentiable on $]0; +\infty[$. We denote $f'$ its derivative, $f''$ its second derivative and $\mathcal{C}_f$ its representative curve in a coordinate system.

1. a. Prove that, for all $x$ belonging to $]0; +\infty[$, we have $f'(x) = \ln(x)$. b. Determine an equation of the tangent line $T$ to the curve $\mathcal{C}_f$ at the point with abscissa $x = \mathrm{e}$. c. Justify that the function $f$ is convex on the interval $]0; +\infty[$. d. Deduce the relative position of the curve $\mathcal{C}_f$ and the tangent line $T$.
   
2. a. Calculate the limit of the function $f$ at 0. b. Prove that the limit of the function $f$ at $+\infty$ is equal to $+\infty$.
   
3. Draw up the table of variations of the function $f$ on the interval $]0; +\infty[$.
   
4. a. Prove that the equation $f(x) = 0$ has a unique solution in the interval $]0; +\infty[$. We denote this solution by $\alpha$. b. Justify that the real number $\alpha$ belongs to the interval $]4{,}3; 4{,}4[$. c. Deduce the sign of the function $f$ on the interval $]0; +\infty[$.
   
5. Consider the following threshold function written in Python: Recall that the \texttt{log} function of the \texttt{math} module (which we assume is imported) denotes the natural logarithm function $\ln$. \begin{verbatim} def seuil(pas) : x=4.3 while x*log(x) - x - 2 < 0: x=x+pas return x \end{verbatim} What is the value returned when calling the function \texttt{seuil(0.01)}? Interpret this result in the context of the exercise.

Exercise 2 — 7 points Themes: Logarithm function and sequence Let $f$ be the function defined on the interval $]0;+\infty[$ by $$f(x) = x\ln(x) + 1$$ We denote $\mathscr{C}_f$ its representative curve in a coordinate system of the plane.

1. Determine the limit of the function $f$ at $0$ as well as its limit at $+\infty$.
   
2. 1. [a.] We admit that $f$ is differentiable on $]0;+\infty[$ and we denote $f'$ its derivative function. Show that for every strictly positive real number $x$: $$f'(x) = 1 + \ln(x).$$
   2. [b.] Deduce the variation table of the function $f$ on $]0;+\infty[$. The exact value of the extremum of $f$ on $]0;+\infty[$ and the limits must be shown.
   3. [c.] Justify that for all $x \in ]0;1[$, $f(x) \in ]0;1[$.
   
   
3. 1. [a.] Determine an equation of the tangent line $(T)$ to the curve $\mathscr{C}_f$ at the point with abscissa $1$.
   2. [b.] Study the convexity of the function $f$ on $]0;+\infty[$.
   3. [c.] Deduce that for every strictly positive real number $x$: $$f(x) \geqslant x$$
   
   
4. The sequence $(u_n)$ is defined by its first term $u_0$ element of the interval $]0;1[$ and for every natural number $n$: $$u_{n+1} = f(u_n)$$
   1. [a.] Prove by induction that for every natural number $n$, we have: $0 < u_n < 1$.
   2. [b.] Deduce from question 3.c. the increasing nature of the sequence $(u_n)$.
   3. [c.] Deduce that the sequence $(u_n)$ is convergent.

Exercise 3 Functions, logarithm function
Let $g$ be the function defined on the interval $]0; +\infty[$ by $$g(x) = 1 + x^{2}[1 - 2\ln(x)]$$ The function $g$ is differentiable on the interval $]0; +\infty[$ and we denote $g'$ its derivative function. We call $\mathscr{C}$ the representative curve of the function $g$ in an orthonormal coordinate system of the plane.

PART A

1. Justify that $g(\mathrm{e})$ is strictly negative.
2. Justify that $\lim_{x \rightarrow +\infty} g(x) = -\infty$.
3. a. Show that, for all $x$ belonging to the interval $]0; +\infty[$, $g'(x) = -4x\ln(x)$. b. Study the direction of variation of the function $g$ on the interval $]0; +\infty[$. c. Show that the equation $g(x) = 0$ admits a unique solution, denoted $\alpha$, on the interval $[1; +\infty[$. d. Give an interval for $\alpha$ with amplitude $10^{-2}$.
4. Deduce from the above the sign of the function $g$ on the interval $[1; +\infty[$.

PART B

1. We admit that, for all $x$ belonging to the interval $[1; \alpha]$, $g''(x) = -4[\ln(x) + 1]$. Justify that the function $g$ is concave on the interval $[1; \alpha]$.
2. In the figure opposite, A and B are points on the curve $\mathscr{C}$ with abscissae respectively 1 and $\alpha$. a. Determine the reduced equation of the line (AB). b. Deduce from this that for all real $x$ belonging to the interval $[1; \alpha]$, $$g(x) \geqslant \frac{-2}{\alpha - 1} x + \frac{2\alpha}{\alpha - 1}.$$

Exercise 3 — 7 points

Topics: Logarithm function, Sequences

Parts $\mathbf { B }$ and $\mathbf { C }$ are independent

We consider the function $f$ defined on $] 0 ; + \infty [$ by $$f ( x ) = x - x \ln x ,$$ where ln denotes the natural logarithm function.

Part A

1. Determine the limit of $f ( x )$ as $x$ tends to 0.
2. Determine the limit of $f ( x )$ as $x$ tends to $+ \infty$.
3. We admit that the function $f$ is differentiable on $] 0 ; + \infty \left[ \right.$ and we denote by $f ^ { \prime }$ its derivative function. a. Prove that, for every real number $x > 0$, we have: $f ^ { \prime } ( x ) = - \ln x$. b. Deduce the variations of the function $f$ on $] 0 ; + \infty [$ and draw its variation table.
4. Solve the equation $f ( x ) = x$ on $] 0$; $+ \infty [$.

Part B

In this part, you may use with profit certain results from Part A. We consider the sequence $(u _ { n })$ defined by: $$\begin{cases} u _ { 0 } & = 0.5 \\ u _ { n + 1 } & = u _ { n } - u _ { n } \ln u _ { n } \text { for every natural number } n , \end{cases}$$ Thus, for every natural number $n$, we have: $u _ { n + 1 } = f \left( u _ { n } \right)$.

1. We recall that the function $f$ is increasing on the interval $[ 0.5 ; 1 ]$. Prove by induction that, for every natural number $n$, we have: $0.5 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 1$.
2. a. Show that the sequence $( u _ { n } )$ is convergent. b. We denote by $\ell$ the limit of the sequence $( u _ { n } )$. Determine the value of $\ell$.

Part C

For any real number $k$, we consider the function $f _ { k }$ defined on $] 0 ; + \infty [$ by: $$f _ { k } ( x ) = k x - x \ln x$$

1. For every real number $k$, show that $f _ { k }$ admits a maximum $y _ { k }$ attained at $x _ { k } = \mathrm { e } ^ { k - 1 }$.
2. Verify that, for every real number $k$, we have: $x _ { k } = y _ { k }$.

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 $]0; e]$: $$[\ln(x)]^2 \geqslant \frac{2}{\mathrm{e}} x - 1$$

Let $f$ be a function defined and differentiable on $\mathbb { R }$. We consider the points $\mathrm { A } ( 1 ; 3 )$ and $\mathrm { B } ( 3 ; 5 )$. We give below $\mathscr { C } _ { f }$ the representative curve of $f$ in an orthogonal coordinate system of the plane, as well as the tangent line (AB) to the curve $\mathscr { C } _ { f }$ at point A.
The three parts of the exercise can be worked on independently.

Part A

1. Determine graphically the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
2. The function $f$ is defined by the expression $f ( x ) = \ln \left( a x ^ { 2 } + 1 \right) + b$, where $a$ and $b$ are positive real numbers. a. Determine the expression of $f ^ { \prime } ( x )$. b. Determine the values of $a$ and $b$ using the previous results.

Part B

It is admitted that the function $f$ is defined on $\mathbb { R }$ by $$f ( x ) = \ln \left( x ^ { 2 } + 1 \right) + 3 - \ln ( 2 )$$

1. Show that $f$ is an even function.
2. Determine the limits of $f$ at $+ \infty$ and at $- \infty$.
3. Determine the expression of $f ^ { \prime } ( x )$. Study the direction of variation of the function $f$ on $\mathbb { R }$. Draw up the table of variations of $f$ showing the exact value of the minimum as well as the limits of $f$ at $- \infty$ and $+ \infty$.
4. Using the table of variations of $f$, give the values of the real number $k$ for which the equation $f ( x ) = k$ admits two solutions.
5. Solve the equation $f ( x ) = 3 + \ln 2$.

Part C

We recall that the function $f$ is defined on $\mathbb{R}$ by $f ( x ) = \ln \left( x ^ { 2 } + 1 \right) + 3 - \ln ( 2 )$.

1. Conjecture, by graphical reading, the abscissas of any inflection points of the curve $\mathscr { C } _ { f }$.
2. Show that, for any real number $x$, we have: $f ^ { \prime \prime } ( x ) = \frac { 2 \left( 1 - x ^ { 2 } \right) } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$.
3. Deduce the largest interval on which the function $f$ is convex.

We admit that the function $f$ from part $\mathbf{A}$ is defined on $\mathbb{R}$ by
$$f(x) = \left(x^{2} - 5x + 6\right)\mathrm{e}^{x}$$
We denote $\mathscr{C}$ the representative curve of the function $f$ in a coordinate system.

1. a. Determine the limit of the function $f$ at $+\infty$. b. Determine the limit of the function $f$ at $-\infty$.
2. Show that, for all real $x$, we have $f^{\prime}(x) = \left(x^{2} - 3x + 1\right)\mathrm{e}^{x}$.
3. Deduce the direction of variation of the function $f$.
4. Determine the reduced equation of the tangent line $(\mathscr{T})$ to the curve $\mathscr{C}$ 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 $f$. We admit that, for all real $x$, we have $f^{\prime\prime}(x) = (x+1)(x-2)\mathrm{e}^{x}$.
5. a. Study the convexity of the function $f$ on $\mathbb{R}$. b. Show that, for all $x$ belonging to the interval $[-1; 2]$, we have $f(x) \leqslant x + 6$.

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: $$f(x) = \mathrm{e}^{3x} - (2x+1)\mathrm{e}^{x}$$
The purpose of this exercise is to study the function $f$ on $\mathbb{R}$.
Part A - Study of an auxiliary function
We define the function $g$ on $\mathbb{R}$ by: $$g(x) = 3\mathrm{e}^{2x} - 2x - 3$$

1. a. Determine the limit of function $g$ at $-\infty$. b. Determine the limit of function $g$ at $+\infty$.
2. a. We admit that function $g$ is differentiable on $\mathbb{R}$, and we denote by $g'$ its derivative function. Prove that for every real number $x$, we have $g'(x) = 6\mathrm{e}^{2x} - 2$. b. Study the sign of the derivative function $g'$ on $\mathbb{R}$. c. Deduce the table of variations of function $g$ on $\mathbb{R}$. Verify that function $g$ has a minimum equal to $\ln(3) - 2$.
3. a. Show that $x = 0$ is a solution of the equation $g(x) = 0$. b. Show that the equation $g(x) = 0$ has a second non-zero solution, denoted $\alpha$, for which you will give an interval of amplitude $10^{-1}$.
4. Deduce from the previous questions the sign of function $g$ on $\mathbb{R}$.

Part B - Study of function $f$

1. Function $f$ is differentiable on $\mathbb{R}$, and we denote by $f'$ its derivative function. Prove that for every real number $x$, we have $f'(x) = \mathrm{e}^{x} g(x)$, where $g$ is the function defined in Part A.
2. Deduce the sign of the derivative function $f'$ and then the variations of function $f$ on $\mathbb{R}$.
3. Why is function $f$ not convex on $\mathbb{R}$? Explain.

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.

Let $f$ be the function defined on $]0; +\infty[$ by $f(x) = x^2 \ln x$. The expression of the derivative function of $f$ is: a. $f'(x) = 2x \ln x$. b. $f'(x) = x(2\ln x + 1)$. c. $f'(x) = 2$. d. $f'(x) = x$.

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 $f$.

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.