The function $f$ is differentiable on the closed interval $[-6, 5]$ and satisfies $f(-2) = 7$. The graph of $f'$, the derivative of $f$, consists of a semicircle and three line segments, as shown in the figure.
(a) Find the values of $f(-6)$ and $f(5)$.
(b) On what intervals is $f$ increasing? Justify your answer.
(c) Find the absolute minimum value of $f$ on the closed interval $[-6, 5]$. Justify your answer.
(d) For each of $f''(-5)$ and $f''(3)$, find the value or explain why it does not exist.

Let $f$ be a differentiable function with $f ( 4 ) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f ^ { \prime }$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above.
(a) Find $f ( 0 )$ and $f ( 5 )$.
(b) Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$. Justify your answer.
(c) Let $g$ be the function defined by $g ( x ) = f ( x ) - x$. On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$ ? Show the analysis that leads to your answer.
(d) For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$. Justify your answer.

Exercise 1 -- Common to all candidates

Part A

Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{1-x}$$

1. Verify that for all real $x$, $f(x) = \mathrm{e} \times \frac{x}{\mathrm{e}^{x}}$.
2. Determine the limit of the function $f$ at $-\infty$.
3. Determine the limit of the function $f$ at $+\infty$. Interpret this limit graphically.
4. Determine the derivative of the function $f$.
5. Study the variations of the function $f$ on $\mathbb{R}$ then draw up the variation table.

Part B

For every non-zero natural number $n$, we consider the functions $g_n$ and $h_n$ defined on $\mathbb{R}$ by: $$g_n(x) = 1 + x + x^2 + \cdots + x^n \quad \text{and} \quad h_n(x) = 1 + 2x + \cdots + nx^{n-1}.$$

1. Verify that, for all real $x$: $(1-x)g_n(x) = 1 - x^{n+1}$.

We then obtain, for all real $x \neq 1$: $g_n(x) = \frac{1 - x^{n+1}}{1-x}$.
2. Compare the functions $h_n$ and $g_n'$, $g_n'$ being the derivative of the function $g_n$.
Deduce that, for all real $x \neq 1$: $h_n(x) = \frac{nx^{n+1} - (n+1)x^n + 1}{(1-x)^2}$.
3. Let $S_n = f(1) + f(2) + \ldots + f(n)$, $f$ being the function defined in Part A.
Using the results from Part B, determine an expression for $S_n$ then its limit as $n$ tends to $+\infty$.

Exercise 2

We consider the function $f$ defined and differentiable on the set $\mathbb { R }$ of real numbers by
$$f ( x ) = x + 1 + \frac { x } { \mathrm { e } ^ { x } }$$
We denote by $\mathscr { C }$ its representative curve in an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).

Part A

1. Let $g$ be the function defined and differentiable on the set $\mathbb { R }$ by $$g ( x ) = 1 - x + \mathrm { e } ^ { x }$$ Draw up, by justifying, the table giving the variations of the function $g$ on $\mathbb { R }$ (the limits of $g$ at the boundaries of its domain are not required). Deduce the sign of $g ( x )$.
2. Determine the limit of $f$ at $- \infty$ then the limit of $f$ at $+ \infty$.
3. We call $f ^ { \prime }$ the derivative of the function $f$ on $\mathbb { R }$. Prove that, for all real $x$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { - x } g ( x )$$
4. Deduce the variation table of the function $f$ on $\mathbb { R }$.
5. Prove that the equation $f ( x ) = 0$ admits a unique real solution $\alpha$ on $\mathbb { R }$. Prove that $- 1 < \alpha < 0$.
6. a. Prove that the line T with equation $y = 2 x + 1$ is tangent to the curve $\mathscr { C }$ at the point with abscissa 0. b. Study the relative position of the curve $\mathscr { C }$ and the line T.

Part B

1. Let H be the function defined and differentiable on $\mathbb { R }$ by $$\mathrm { H } ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }.$$ Prove that H is a primitive on $\mathbb { R }$ of the function $h$ defined by $h ( x ) = x \mathrm { e } ^ { - x }$.
2. We denote by $\mathscr { D }$ the domain bounded by the curve $\mathscr { C }$, the line T and the lines with equations $x = 1$ and $x = 3$. Calculate, in square units, the area of the domain $\mathscr { D }$.

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

Part A

1. We denote by $f'$ the derivative function of the function $f$ on the interval $[0; +\infty[$. For every real number $x$ in the interval $[0; +\infty[$, calculate $f'(x)$. Deduce the variations of the function $f$ on the interval $[0; +\infty[$.
2. Determine the limit of the function $f$ at $+\infty$. What graphical interpretation can be made of this result?

Part B

Let $\mathscr{A}$ be the function defined on the interval $[0; +\infty[$ as follows: for every real number $t$ in the interval $[0; +\infty[$, $\mathscr{A}(t)$ is the area, in square units, of the region bounded by the $x$-axis, the curve $\mathscr{C}$ and the lines with equations $x = 0$ and $x = t$.

1. Determine the direction of variation of the function $\mathscr{A}$.
2. We admit that the area of the region bounded by the curve $\mathscr{C}$ and the $x$-axis is equal to 1 square unit. What can we deduce about the function $\mathscr{A}$?
3. We seek to prove the existence of a real number $\alpha$ such that the line with equation $x = \alpha$ divides the region between the $x$-axis and the curve $\mathscr{C}$ into two parts of equal area, and to find an approximate value of this real number. a) Prove that the equation $\mathscr{A}(t) = \frac{1}{2}$ has a unique solution on the interval $[0; +\infty[$ b) On the graph provided in the appendix (to be returned with the answer sheet) are drawn the curve $\mathscr{C}$, as well as the curve $\Gamma$ representing the function $\mathscr{A}$. On the graph in the appendix, identify the curves $\mathscr{C}$ and $\Gamma$, then draw the line with equation $y = \frac{1}{2}$. Deduce an approximate value of the real number $\alpha$. Shade the region corresponding to $\mathscr{A}(\alpha)$.
4. We define the function $g$ on the interval $[0; +\infty[$ by $$g(x) = (x + 1)\mathrm{e}^{-x}$$ a) We denote by $g'$ the derivative function of the function $g$ on the interval $[0; +\infty[$. For every real number $x$ in the interval $[0; +\infty[$, calculate $g'(x)$. b) Deduce, for every real number $t$ in the interval $[0; +\infty[$, an expression for $\mathscr{A}(t)$. c) Calculate an approximate value to $10^{-2}$ of $\mathscr{A}(6)$.

Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x - \ln\left(x^{2} + 1\right).$$

1. Solve in $\mathbb{R}$ the equation: $f(x) = x$.
2. Justify all elements of the variation table below except for the limit of the function $f$ at $+\infty$ which is admitted.
   

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

   
   
3. Show that, for every real $x$ belonging to $[0;1]$, $f(x)$ belongs to $[0;1]$.
4. Consider the following algorithm:
   

   Variables	$N$ and $A$ natural integers;
   Input	Enter the value of $A$
   Processing	\begin{tabular}{ l } $N$ takes the value 0
   While $N - \ln\left(N^{2} + 1\right) < A$
   $N$ takes the value $N + 1$
   End while

   \hline Output & Display $N$ \hline \end{tabular}
   a. What does this algorithm do? b. Determine the value $N$ provided by the algorithm when the value entered for $A$ is 100.

Let $f$ and $g$ be the functions defined on $] 0 ; + \infty [$ by
$$f ( x ) = \mathrm { e } ^ { - x } \quad \text { and } \quad g ( x ) = \frac { 1 } { x ^ { 2 } } \mathrm { e } ^ { - \frac { 1 } { x } } .$$
We admit that $f$ and $g$ are differentiable on $] 0 ; + \infty [$. We denote $f ^ { \prime }$ and $g ^ { \prime }$ their respective derivative functions.

Part A - Graphical conjectures

In each of the questions in this part, no explanation is required.

1. Conjecture graphically a solution to the equation $f ( x ) = g ( x )$ on $] 0 ; + \infty [$.
2. Conjecture graphically a solution to the equation $g ^ { \prime } ( x ) = 0$ on $] 0 ; + \infty [$.

Part B - Study of the function $g$

1. Calculate the limit of $g ( x )$ as $x$ tends to $+ \infty$.
2. We admit that the function $g$ is strictly positive on $] 0 ; + \infty [$.

Let $h$ be the function defined on $] 0 ; + \infty [$ by $h ( x ) = \ln ( g ( x ) )$. a. Prove that, for every strictly positive real number $x$,
$$h ( x ) = \frac { - 1 - 2 x \ln x } { x } .$$
b. Calculate the limit of $h ( x )$ as $x$ tends to 0. c. Deduce the limit of $g ( x )$ as $x$ tends to 0.
3. Prove that, for every strictly positive real number $x$,
$$g ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { - \frac { 1 } { x } } ( 1 - 2 x ) } { x ^ { 4 } } .$$

1. Deduce the variations of the function $g$ on $] 0 ; + \infty [$.

Part C - Area of the two regions between the curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$

1. Prove that point A with coordinates $( 1 ; \mathrm { e } ^ { - 1 } )$ is an intersection point of $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$. We admit that this point is the unique intersection point of $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$, and that $\mathscr { C } _ { f }$ is above $\mathscr { C } _ { g }$ on the interval $] 0 ; 1 [$ and below on the interval $] 1 ; + \infty [$.
2. Let $a$ and $b$ be two strictly positive real numbers. Prove that

$$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) \mathrm { d } x = \mathrm { e } ^ { - a } + \mathrm { e } ^ { - \frac { 1 } { a } } - \mathrm { e } ^ { - b } - \mathrm { e } ^ { - \frac { 1 } { b } } .$$

1. Prove that

$$\lim _ { a \rightarrow 0 } \int _ { a } ^ { 1 } ( f ( x ) - g ( x ) ) \mathrm { d } x = 1 - 2 \mathrm { e } ^ { - 1 } .$$

1. We admit that

$$\lim _ { a \rightarrow 0 } \int _ { a } ^ { 1 } ( f ( x ) - g ( x ) ) \mathrm { d } x = \lim _ { b \rightarrow + \infty } \int _ { 1 } ^ { b } ( g ( x ) - f ( x ) ) \mathrm { d } x .$$
Give a graphical interpretation of this equality.

Consider the function $f$ defined on $\mathbb { R }$ by $$f ( x ) = x \mathrm { e } ^ { - x ^ { 2 } + 1 }$$ Let ( $\mathscr { C }$ ) denote the representative curve of $f$ in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).

1. a. Show that for every real $x$, $f ( x ) = \frac { \mathrm { e } } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$. b. Deduce the limit of $f ( x )$ as $x$ tends to $+ \infty$.
2. For every real $x$, consider the points $M$ and $N$ on the curve $( \mathscr { C } )$ with abscissae $x$ and $- x$ respectively. a. Show that the point O is the midpoint of the segment $[ M N ]$. b. What can be deduced about the curve $( \mathscr { C } )$ ?
3. Study the variations of the function $f$ on the interval $[ 0 ; + \infty [$.
4. a. Show that the equation $f ( x ) = 0.5$ admits on $[ 0 ; + \infty [$ exactly two solutions denoted $\alpha$ and $\beta$ (with $\alpha < \beta$ ). b. Deduce the solutions on $[ 0 ; + \infty [$ of the inequality $f ( x ) \geqslant 0.5$. c. Give an approximate value to $10 ^ { - 2 }$ of $\alpha$ and $\beta$.
5. Let $A$ be a strictly positive real number. Set $I _ { A } = \int _ { 0 } ^ { A } f ( x ) \mathrm { d } x$. a. Justify that $I _ { A } = \frac { 1 } { 2 } \left( \mathrm { e } - \mathrm { e } ^ { - A ^ { 2 } + 1 } \right)$. b. Calculate the limit of $I _ { A }$ as $A$ tends to $+ \infty$.
6. As illustrated in the graph below, we are interested in the shaded part of the plane which is bounded by:
   • the curve $( \mathscr { C } )$ on $\mathbb { R }$ and the curve $\left( \mathscr { C } ^ { \prime } \right)$ symmetric to $( \mathscr { C } )$ with respect to the x-axis;
   • the circle with centre $\Omega \left( \frac { \sqrt { 2 } } { 2 } ; 0 \right)$ and radius 0.5 and its symmetric with respect to the y-axis.
   It is admitted that the disk with centre $\Omega \left( \frac { \sqrt { 2 } } { 2 } ; 0 \right)$ and radius 0.5 and its symmetric with respect to the y-axis are situated entirely between the curve ( $\mathscr { C }$ ) and the curve ( $\mathscr { C } ^ { \prime }$ ). Determine an approximate value in square units to the nearest hundredth of the area of this shaded part of the plane.

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

1. By noting that for all $x$ in $[ 0 ; + \infty [$, we have

$$f ( x ) = \frac { x } { \mathrm { e } ^ { x } }$$ prove that the curve $\mathscr { C } _ { f }$ has an asymptote at $+ \infty$ for which you will give an equation.
2. Prove that for all real $x$ belonging to $[ 0 ; + \infty [$ : $$f ^ { \prime } ( x ) = ( 1 - x ) \mathrm { e } ^ { - x }$$

1. Draw up the table of variations of $f$ on $[ 0 ; + \infty [$, on which you will show the values at the boundaries as well as the exact value of the extremum.
2. Determine, on the interval $[ 0 ; + \infty [$, the number of solutions of the equation

$$f ( x ) = \frac { 367 } { 1000 }$$

1. We admit that for all $x$ belonging to $[ 0 ; + \infty [$ :

$$f ^ { \prime \prime } ( x ) = \mathrm { e } ^ { - x } ( x - 2 )$$ Study the convexity of the function $f$ on the interval $[ 0 ; + \infty [$. 6. Let $a$ be a real number belonging to $[ 0 ; + \infty [$ and A the point of the curve $\mathscr { C } _ { f }$ with abscissa $a$. We denote $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at A. We denote $\mathrm { H } _ { a }$ the point of intersection of the line $T _ { a }$ and the ordinate axis. We denote $g ( a )$ the ordinate of $\mathrm { H } _ { a }$. a. Prove that a reduced equation of the tangent $T _ { a }$ is: $$y = \left[ ( 1 - a ) \mathrm { e } ^ { - a } \right] x + a ^ { 2 } \mathrm { e } ^ { - a }$$ b. Deduce the expression of $g ( a )$. c. Prove that $g ( a )$ is maximum when A is an inflection point of the curve $\mathscr { C } _ { f }$.

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 $$f(x) = x^2 - 8\ln(x)$$ 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 $]0; +\infty[$.

Part A Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(x + \frac{1}{2}\right)e^{-x} + x.$$

1. Determine the limits of $f$ at $-\infty$ and at $+\infty$.
2. It is admitted that $f$ is twice differentiable on $\mathbb{R}$. a. Prove that, for all $x \in \mathbb{R}$, $$f''(x) = \left(x - \frac{3}{2}\right)e^{-x}.$$ b. Deduce the variations and the minimum of the function $f'$ on $\mathbb{R}$. c. Justify that for all $x \in \mathbb{R}$, $f'(x) > 0$. d. Deduce that the equation $f(x) = 0$ admits a unique solution on $\mathbb{R}$. e. Give a value rounded to $10^{-3}$ of this solution.

Part B Consider a function $h$, defined and differentiable on $\mathbb{R}$, having an expression of the form $$h(x) = (ax + b)e^{-x} + x \text{, where } a \text{ and } b \text{ are two real numbers.}$$ In an orthonormal coordinate system shown below are:

• the representative curve $\mathscr{C}_h$ of the function $h$;
• the points A and B with coordinates respectively $(-2; -2.5)$ and $(2; 3.5)$.

1. Conjecture, with the precision allowed by the graph, the abscissae of any inflection points of the representative curve of the function $h$.
2. Knowing that the function $h$ admits on $\mathbb{R}$ a second derivative with expression $$h''(x) = -\frac{3}{2}e^{-x} + xe^{-x}.$$ validate or not the previous conjecture.
3. Determine an equation of the line (AB).
4. Knowing that the line (AB) is tangent to the representative curve of the function $h$ at the point with abscissa 0, deduce the values of $a$ and $b$.

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 $x$: $$\ln ( x ) \geqslant 1 - \frac { 1 } { x } .$$

Part A We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \frac{6}{1 + 5e^{-x}}$$ We have represented on the diagram below the representative curve $\mathscr{C}_f$ of the function $f$.

1. Show that point A with coordinates $(\ln 5 ; 3)$ belongs to the curve $\mathscr{C}_f$.
2. Show that the line with equation $y = 6$ is an asymptote to the curve $\mathscr{C}_f$.
3. a. We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. Show that for every real $x$, we have: $$f'(x) = \frac{30\mathrm{e}^{-x}}{\left(1 + 5\mathrm{e}^{-x}\right)^2}.$$ b. Deduce the complete table of variations of $f$ on $\mathbb{R}$.
4. We admit that:
   • $f$ is twice differentiable on $\mathbb{R}$, we denote $f''$ its second derivative;
   • for every real $x$,
   $$f''(x) = \frac{30\mathrm{e}^{-x}\left(5\mathrm{e}^{-x} - 1\right)}{\left(1 + 5\mathrm{e}^{-x}\right)^3}.$$ a. Study the convexity of $f$ on $\mathbb{R}$. In particular, we will show that the curve $\mathscr{C}_f$ admits an inflection point. b. Justify that for every real $x$ belonging to $]-\infty ; \ln 5]$, we have: $f(x) \geqslant \frac{5}{6}x + 1$.
5. We consider a function $F_k$ defined on $\mathbb{R}$ by $F_k(x) = k\ln\left(\mathrm{e}^x + 5\right)$, where $k$ is a real constant. a. Determine the value of the real $k$ so that $F_k$ is a primitive of $f$ on $\mathbb{R}$. b. Deduce that the area, in square units, of the domain bounded by the curve $\mathscr{C}_f$, the $x$-axis, the $y$-axis and the line with equation $x = \ln 5$ is equal to $6\ln\left(\frac{5}{3}\right)$.

Part B The objective of this part is to study the following differential equation: $$(E) \quad y' = y - \frac{1}{6}y^2.$$ We recall that a solution of equation $(E)$ is a function $u$ defined and differentiable on $\mathbb{R}$ such that for every real $x$, we have: $$u'(x) = u(x) - \frac{1}{6}[u(x)]^2.$$

1. Show that the function $f$ defined in part A is a solution of the differential equation $(E)$.
2. Solve the differential equation $y' = -y + \frac{1}{6}$.
3. We denote by $g$ a function differentiable on $\mathbb{R}$ that does not vanish. We denote by $h$ the function defined on $\mathbb{R}$ by $h(x) = \frac{1}{g(x)}$. We admit that $h$ is differentiable on $\mathbb{R}$. We denote $g'$ and $h'$ the derivative functions of $g$ and $h$. a. Show that if $h$ is a solution of the differential equation $y' = -y + \frac{1}{6}$, then $g$ is a solution of the differential equation $y' = y - \frac{1}{6}y^2$. b. For every positive real $m$, we consider the functions $g_m$ defined on $\mathbb{R}$ by: $$g_m(x) = \frac{6}{1 + 6m\mathrm{e}^{-x}}.$$ Show that for every positive real $m$, the function $g_m$ is a solution of the differential equation $(E): \quad y' = y - \frac{1}{6}y^2$.

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 $\mathscr { A }$.

Consider a function $f$ defined and twice differentiable on $]-2; +\infty[$. We denote by $\mathscr{C}_f$ its representative curve in an orthogonal coordinate system of the plane, $f'$ its derivative and $f''$ its second derivative. The curve $\mathscr{C}_f$ and its tangent $T$ at point B with abscissa $-1$ are drawn. It is specified that the line $T$ passes through the point $\mathrm{A}(0; -1)$.
Part A: exploitation of the graph.
Using the graph, answer the questions below.

1. Specify $f(-1)$ and $f'(-1)$.
2. Is the function $f$ convex on its domain of definition? Justify.
3. Conjecture the number of solutions of the equation $f(x) = 0$ and give a value rounded to $10^{-1}$ near a solution.

Part B: study of the function $f$
Consider that the function $f$ is defined on $]-2; +\infty[$ by: $$f(x) = x^2 + 2x - 1 + \ln(x+2),$$ where ln denotes the natural logarithm function.

1. Determine by calculation the limit of the function $f$ at $-2$. Interpret this result graphically.

It is admitted that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.

1. Show that for all $x > -2$, $\quad f'(x) = \frac{2x^2 + 6x + 5}{x+2}$.
2. Study the variations of the function $f$ on $]-2; +\infty[$ then draw up its complete variation table.
3. Show that the equation $f(x) = 0$ admits a unique solution $\alpha$ on $]-2; +\infty[$ and give a value of $\alpha$ rounded to $10^{-2}$ near.
4. Deduce the sign of $f(x)$ on $]-2; +\infty[$.
5. Show that $\mathscr{C}_f$ admits a unique inflection point and determine its abscissa.

Part C: a minimum distance.
Let $g$ be the function defined on $]-2; +\infty[$ by $\quad g(x) = \ln(x+2)$. We denote by $\mathscr{C}_g$ its representative curve in an orthonormal coordinate system $(O; I, J)$. Let $M$ be a point of $\mathscr{C}_g$ with abscissa $x$. The purpose of this part is to determine for which value of $x$ the distance $JM$ is minimal. Consider the function $h$ defined on $]-2; +\infty[$ by $\quad h(x) = JM^2$.

1. Justify that for all $x > -2$, we have: $\quad h(x) = x^2 + [\ln(x+2) - 1]^2$.
2. It is admitted that the function $h$ is differentiable on $]-2; +\infty[$ and we denote by $h'$ its derivative function. It is also admitted that for all real $x > -2$, $$h'(x) = \frac{2f(x)}{x+2}$$ where $f$ is the function studied in part B. a. Draw up the variation table of $h$ on $]-2; +\infty[$. The limits are not required. b. Deduce that the value of $x$ for which the distance $JM$ is minimal is $\alpha$ where $\alpha$ is the real number defined in question 4 of part B.
3. We will denote by $M_\alpha$ the point of $\mathscr{C}_g$ with abscissa $\alpha$. a. Show that $\ln(\alpha + 2) = 1 - 2\alpha - \alpha^2$. b. Deduce that the tangent to $\mathscr{C}_g$ at point $M_\alpha$ and the line $(JM_\alpha)$ are perpendicular. One may use the fact that, in an orthonormal coordinate system, two lines are perpendicular when the product of their slopes is equal to $-1$.

We consider the function $f$ defined on the interval $] 0 ; + \infty [$ by $$f ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } } { 2 \sqrt { x } }$$ and we call $\mathscr { C } _ { f }$ 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: $$f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( \sqrt { x } - 1 ) } { 4 x \sqrt { x } } .$$
2. a. Determine the limit of the function $f$ at 0. b. Interpret this result graphically.
3. 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 $f$ showing the limits at the boundaries of the domain of definition. c. Show that the equation $f ( x ) = 2$ has a unique solution on the interval $\left[ 1 ; + \infty \left[ \right. \right.$ and give an approximate value to $10 ^ { - 1 }$ of this solution.
4. We set $I = \int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x$. a. Calculate $I$. b. Interpret the result graphically.
5. 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 $] 0 ; + \infty [$.

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 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?

For a positive number $t$, the function $f ( x )$ defined on the interval $[ 1 , \infty )$ is $$f ( x ) = \begin{cases} \ln x & ( 1 \leq x < e ) \\ - t + \ln x & ( x \geq e ) \end{cases}$$ Among linear functions $g ( x )$ satisfying the following condition, let $h ( t )$ be the minimum value of the slope of the line $y = g ( x )$.
For all real numbers $x \geq 1$, $( x - e ) \{ g ( x ) - f ( x ) \} \geq 0$.
For a differentiable function $h ( t )$, a positive number $a$ satisfies $h ( a ) = \frac { 1 } { e + 2 }$. What is the value of $h ^ { \prime } \left( \frac { 1 } { 2 e } \right) \times h ^ { \prime } ( a )$? [4 points]
(1) $\frac { 1 } { ( e + 1 ) ^ { 2 } }$
(2) $\frac { 1 } { e ( e + 1 ) }$
(3) $\frac { 1 } { e ^ { 2 } }$
(4) $\frac { 1 } { ( e - 1 ) ( e + 1 ) }$
(5) $\frac { 1 } { e ( e - 1 ) }$

21. Given the function $f ( x ) = \frac { a x } { ( x + r ) ^ { 2 } } ( a > 0 , r > 0 )$
(1) Find the domain of $f ( x )$ and discuss the monotonicity of $f ( x )$;
(2) If $\frac { a } { r } = 400$, find the extreme values of $f ( x )$ on $( 0 , + \infty )$.

(12 points)
Given the function $f(x) = e^x(e^x - a) - a^2x$.
(1) Discuss the monotonicity of $f(x)$;
(2) If $f(x) \geq 0$, find the range of values for $a$.

Study the variations of the function $t \mapsto t\ln(t)$ on $\mathbf{R}_+^*$. Verify that we can extend the function by continuity at $0$.

Let the function $f : [0,1] \rightarrow \mathbb{R}$ be defined as $$f(x) = \max\left\{\frac{|x - y|}{x + y + 1} : 0 \leq y \leq 1\right\} \text{ for } 0 \leq x \leq 1$$ Then which of the following statements is correct?
(A) $f$ is strictly increasing on $\left[0, \frac{1}{2}\right]$ and strictly decreasing on $\left[\frac{1}{2}, 1\right]$.
(B) $f$ is strictly decreasing on $\left[0, \frac{1}{2}\right]$ and strictly increasing on $\left[\frac{1}{2}, 1\right]$.
(C) $f$ is strictly increasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly decreasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.
(D) $f$ is strictly decreasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly increasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.

Let $f : ( - 1 , \infty ) \rightarrow R$ be defined by $f ( 0 ) = 1$ and $f ( x ) = \frac { 1 } { x } \log _ { e } ( 1 + x ) , x \neq 0$. Then the function $f$
(1) Decreases in $( - 1,0 )$ and increases in $( 0 , \infty )$
(2) Increases in $( - 1 , \infty )$
(3) Increases in $( - 1,0 )$ and decreases in $( 0 , \infty )$
(4) Decreases in $( - 1 , \infty )$