Let $f$ be the function defined for $x > 0$, with $f(e) = 2$ and $f'$, the first derivative of $f$, given by $f'(x) = x^2 \ln x$.
(a) Write an equation for the line tangent to the graph of $f$ at the point $(e, 2)$.
(b) Is the graph of $f$ concave up or concave down on the interval $1 < x < 3$? Give a reason for your answer.
(c) Use antidifferentiation to find $f(x)$.

Exercise 1 -- Common to all candidates

Part A

We consider the function $h$ defined on the interval $[ 0 ; + \infty [$ by: $$h ( x ) = x \mathrm { e } ^ { - x }$$

1. Determine the limit of the function $h$ at $+ \infty$.
2. Study the variations of the function $h$ on the interval $[ 0 ; + \infty [$ and draw up its table of variations.
3. The objective of this question is to determine a primitive of the function $h$. a. Verify that for every real number $x$ belonging to the interval $[ 0 ; + \infty [$, we have: $$h ( x ) = \mathrm { e } ^ { - x } - h ^ { \prime } ( x )$$ where $h ^ { \prime }$ denotes the derivative function of $h$. b. Determine a primitive on the interval $[ 0 ; + \infty [$ of the function $x \longmapsto \mathrm { e } ^ { - x }$. c. Deduce from the two previous questions a primitive of the function $h$ on the interval $[ 0 ; + \infty [$.

Part B

We define the functions $f$ and $g$ on the interval $[ 0 ; + \infty [$ by: $$f ( x ) = x \mathrm { e } ^ { - x } + \ln ( x + 1 ) \quad \text { and } \quad g ( x ) = \ln ( x + 1 )$$ We denote $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ the respective graphical representations of the functions $f$ and $g$ in an orthonormal coordinate system.

1. For a real number $x$ belonging to the interval $[ 0 ; + \infty [$, we call $M$ the point with coordinates $( x ; f ( x ) )$ and $N$ the point with coordinates $( x ; g ( x ) )$: $M$ and $N$ are therefore the points with abscissa $x$ belonging respectively to the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$. a. Determine the value of $x$ for which the distance $MN$ is maximum and give this maximum distance. b. Place on the graph provided in the appendix the points $M$ and $N$ corresponding to the maximum value of $MN$.
2. Let $\lambda$ be a real number belonging to the interval $[ 0 ; + \infty [$. We denote $D _ { \lambda }$ the region of the plane bounded by the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ and by the lines with equations $x = 0$ and $x = \lambda$. a. Shade the region $D _ { \lambda }$ corresponding to the value $\lambda$ proposed on the graph in the appendix. b. We denote $A _ { \lambda }$ the area of the region $D _ { \lambda }$, expressed in square units. Prove that: $$A _ { \lambda } = 1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } .$$ c. Calculate the limit of $A _ { \lambda }$ as $\lambda$ tends to $+ \infty$ and interpret the result.
3. We consider the following algorithm: \begin{verbatim} Variables: $\lambda$ is a positive real number $S$ is a real number strictly between 0 and 1. Initialization: Input $S$ $\lambda$ takes the value 0 Processing: While $1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } < S$ do $\lambda$ takes the value $\lambda + 1$ End While Output: Display $\lambda$ \end{verbatim} a. What value does this algorithm display if we input the value $S = 0.8$? b. What is the role of this algorithm?

The flow of water from a tap has a constant and moderate flow rate.
We are particularly interested in a part of the flow profile represented in appendix 1 by the curve $C$ in an orthonormal coordinate system.

Part A

We consider that the curve $C$ given in appendix 1 is the graphical representation of a function $f$ differentiable on the interval $] 0 ; 1 ]$ which respects the following three conditions:
$$( H ) : f ( 1 ) = 0 \quad f ^ { \prime } ( 1 ) = 0.25 \quad \text { and } \lim _ { \substack { x \rightarrow 0 \\ x > 0 } } f ( x ) = - \infty .$$

1. Can the function $f$ be a polynomial function of degree two? Why?
2. Let $g$ be the function defined on the interval $]0 ; 1]$ by $g ( x ) = k \ln x$. a. Determine the real number $k$ so that the function $g$ respects the three conditions $( H )$. b. Does the representative curve of the function $g$ coincide with the curve $C$ ? Why?
3. Let $h$ be the function defined on the interval $]0; 1]$ by $h ( x ) = \frac { a } { x ^ { 4 } } + b x$ where $a$ and $b$ are real numbers. Determine $a$ and $b$ so that the function $h$ respects the three conditions ( $H$ ).

Part B

We admit in this part that the curve $C$ is the graphical representation of a function $f$ continuous, strictly increasing, defined and differentiable on the interval $] 0 ; 1 ]$ with expression:
$$f ( x ) = \frac { 1 } { 20 } \left( x - \frac { 1 } { x ^ { 4 } } \right)$$

1. Justify that the equation $f ( x ) = - 5$ admits on the interval $] 0 ; 1 ]$ a unique solution which will be denoted $\alpha$. Determine an approximate value of $\alpha$ to $10 ^ { - 2 }$ near.
2. It is admitted that the volume of water in $\mathrm { cm } ^ { 3 }$, contained in the first 5 centimetres of the flow, is given by the formula: $V = \int _ { \alpha } ^ { 1 } \pi x ^ { 2 } f ^ { \prime } ( x ) \mathrm { d } x$. a. Let $u$ be the function differentiable on $] 0; 1]$ defined by $u ( x ) = \frac { 1 } { 2 x ^ { 2 } }$. Determine its derivative function. b. Determine the exact value of $V$. Using the approximate value of $\alpha$ obtained in question 1, give an approximate value of $V$.

In this part, we consider that the function $f$, defined and twice differentiable on $[0; +\infty[$, is defined by
$$f(x) = (4x - 2)\mathrm{e}^{-x + 1}.$$
We will denote respectively $f'$ and $f''$ the derivative and second derivative of the function $f$.

1. Study of the function $f$ a. Show that $f'(x) = (-4x + 6)\mathrm{e}^{-x + 1}$. b. Use this result to determine the complete table of variations of the function $f$ on $[0; +\infty[$. It is admitted that $\lim_{x \rightarrow +\infty} f(x) = 0$. c. Study the convexity of the function $f$ and specify the abscissa of any possible inflection point of the representative curve of $f$.
2. We consider a function $F$ defined on $[0; +\infty[$ by $F(x) = (ax + b)\mathrm{e}^{-x + 1}$, where $a$ and $b$ are two real numbers. a. Determine the values of the real numbers $a$ and $b$ such that the function $F$ is a primitive of the function $f$ on $[0; +\infty[$. b. It is admitted that $F(x) = (-4x - 2)\mathrm{e}^{-x + 1}$ is a primitive of the function $f$ on $[0; +\infty[$. Deduce the exact value, then an approximate value to $10^{-2}$ near, of the integral $$I = \int_{\frac{3}{2}}^{8} f(x)\mathrm{d}x.$$
3. A municipality has decided to build a freestyle scooter track. The profile of this track is given by the representative curve of the function $f$ on the interval $[\frac{3}{2}; 8]$. The unit of length is the meter. a. Give an approximate value to the nearest cm of the height of the starting point D. b. The municipality has organized a graffiti competition to decorate the wall profile of the track. The selected artist plans to cover approximately $75\%$ of the wall surface. Knowing that a 150 mL aerosol can covers a surface of $0.8\mathrm{~m}^2$, determine the number of cans she will need to use to create this work.

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

PART A

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

PART B

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

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

In a mathematics class where the topic of integration by parts is being taught, Teacher Ebru writes on the board
$$\int u d v = u v - \int v d u$$
rule. Then, in solving a problem, Mehmet chooses functions $f ( x )$ and $g ( x )$ in place of u and v respectively and applies this rule to obtain
$$\int f ( x ) g ^ { \prime } ( x ) d x = \frac { f ( x ) } { x } - \int \frac { 2 } { x ^ { 2 } } d x$$
equality.
Given that $\mathbf { f } ( \mathbf { 1 } ) = \mathbf { 2 }$, what is the value of $\mathbf { f } ( \mathbf { e } )$?
A) 2 B) 4 C) 6 D) 8 E) 10