Part A

In the plane with an orthonormal coordinate system, we denote by $\mathscr { C } _ { 1 }$ the curve representing the function $f _ { 1 }$ defined on $\mathbb { R }$ by: $$f _ { 1 } ( x ) = x + \mathrm { e } ^ { - x } .$$

1. Justify that $\mathscr { C } _ { 1 }$ passes through point A with coordinates $( 0 ; 1 )$.
2. Determine the variation table of the function $f _ { 1 }$. Specify the limits of $f _ { 1 }$ at $+ \infty$ and at $- \infty$.

Part B

The purpose of this part is to study the sequence $\left( I _ { n } \right)$ defined on $\mathbb { N }$ by: $$I _ { n } = \int _ { 0 } ^ { 1 } \left( x + \mathrm { e } ^ { - n x } \right) \mathrm { d } x .$$

1. In the plane with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ), for every natural integer $n$, we denote by $\mathscr { C } _ { n }$ the curve representing the function $f _ { n }$ defined on $\mathbb { R }$ by $$f _ { n } ( x ) = x + \mathrm { e } ^ { - n x } .$$ a. Give a geometric interpretation of the integral $I _ { n }$. b. Using this interpretation, formulate a conjecture about the direction of variation of the sequence ( $I _ { n }$ ) and its possible limit. Specify the elements on which you base your conjecture.
2. Prove that for every natural integer $n$ greater than or equal to 1, $$I _ { n + 1 } - I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - ( n + 1 ) x } \left( 1 - \mathrm { e } ^ { x } \right) \mathrm { d } x$$ Deduce the sign of $I _ { n + 1 } - I _ { n }$ and then prove that the sequence ( $I _ { n }$ ) is convergent.
3. Determine the expression of $I _ { n }$ as a function of $n$ and determine the limit of the sequence $\left( I _ { n } \right)$.

Let $f$ be the function defined and differentiable on the interval $[ 0 ; + \infty [$ such that:
$$f ( x ) = \frac { x } { \mathrm { e } ^ { x } - x }$$
It is admitted that the function $f$ is positive on the interval $[ 0 ; + \infty [$. We denote by $\mathscr { C }$ the representative curve of the function $f$ in an orthogonal coordinate system of the plane. The curve $\mathscr { C }$ is represented in the appendix, to be returned with the answer sheet.

Part A

Let the sequence $\left( I _ { n } \right)$ be defined for every natural integer $n$ by $I _ { n } = \int _ { 0 } ^ { n } f ( x ) \mathrm { d } x$. We will not seek to calculate the exact value of $I _ { n }$ as a function of $n$.

1. Show that the sequence ( $I _ { n }$ ) is increasing.
2. It is admitted that for every real $x$ in the interval $\left[ 0 ; + \infty \left[ , \mathrm { e } ^ { x } - x \geqslant \frac { \mathrm { e } ^ { x } } { 2 } \right. \right.$. a. Show that, for every natural integer $n , I _ { n } \leqslant \int _ { 0 } ^ { n } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x$. b. Let $H$ be the function defined and differentiable on the interval $[ 0 ; + \infty [$ such that: $$H ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }$$ Determine the derivative function $H ^ { \prime }$ of the function $H$. c. Deduce that, for every natural integer $n , I _ { n } \leqslant 2$.
3. Show that the sequence ( $I _ { n }$ ) is convergent. The value of its limit is not required.

Part B

Consider the following algorithm in which the variables are

• $K$ and $i$ natural integers, $K$ being non-zero;
• $A , x$ and $h$ real numbers.

Input:	Enter $K$ non-zero natural integer
Initialization	\begin{tabular}{l} Assign to $A$ the value 0
Assign to $x$ the value 0
Assign to $h$ the value $\frac { 1 } { K }$

\hline Processing &

For $i$ ranging from 1 to $K$

Assign to $A$ the value $A + h \times f ( x )$

Assign to $x$ the value $x + h$

End For

\hline Output & Display $A$ \hline \end{tabular}

1. Reproduce and complete the following table by running this algorithm for $K = 4$. The successive values of $A$ will be rounded to the nearest thousandth.
   

   $i$	$A$	$x$
   1
   2
   3
   4

   
   
2. By illustrating it on the appendix to be returned with the answer sheet, give a graphical interpretation of the result displayed by this algorithm for $K = 8$.
3. What does the algorithm give when $K$ becomes large?

We consider, for every integer $n > 0$, the functions $f_{n}$ defined on the interval $[1; 5]$ by: $$f_{n}(x) = \frac{\ln x}{x^{n}}$$ For every integer $n > 0$, we denote by $\mathscr{C}_{n}$ the representative curve of the function $f_{n}$ in an orthogonal reference frame.

1. Show that, for every integer $n > 0$ and every real $x$ in the interval $[1; 5]$: $$f_{n}^{\prime}(x) = \frac{1 - n\ln(x)}{x^{n+1}}$$
2. For every integer $n > 0$, we admit that the function $f_{n}$ has a maximum on the interval $[1; 5]$. We denote by $A_{n}$ the point of the curve $\mathscr{C}_{n}$ having as ordinate this maximum. Show that all points $A_{n}$ belong to the same curve $\Gamma$ with equation $$y = \frac{1}{\mathrm{e}} \ln(x)$$
3. a. Show that, for every integer $n > 1$ and every real $x$ in the interval $[1; 5]$: $$0 \leqslant \frac{\ln(x)}{x^{n}} \leqslant \frac{\ln(5)}{x^{n}}$$ b. Show that for every integer $n > 1$: $$\int_{1}^{5} \frac{1}{x^{n}} \mathrm{~d}x = \frac{1}{n-1}\left(1 - \frac{1}{5^{n-1}}\right)$$ c. For every integer $n > 0$, we are interested in the area, expressed in square units, of the surface under the curve $f_{n}$, that is the area of the region of the plane bounded by the lines with equations $x = 1$, $x = 5$, $y = 0$ and the curve $\mathscr{C}_{n}$. Determine the limiting value of this area as $n$ tends to $+\infty$.

Prove that $\int _ { 1 } ^ { b } a ^ { \log _ { b } x } d x > \ln b$ where $a , b > 0 , b \neq 1$.

Consider the function $f(x) = x^{\cos(x) + \sin(x)}$ defined for $x \geq 0$.
(a) Prove that
$$0.4 \leq \int_{0}^{1} f(x)\, dx \leq 0.5$$
(b) Suppose the graph of $f(x)$ is being traced on a computer screen with the uniform speed of 1 cm per second (i.e., this is how fast the length of the curve is increasing). Show that at the moment the point corresponding to $x = 1$ is being drawn, the $x$ coordinate is increasing at the rate of
$$\frac{1}{\sqrt{2 + \sin(2)}} \text{ cm per second.}$$

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Show that the sequence $$\left[\int_0^1 |f(x)|^n\,\mathrm{d}x\right]^{\frac{1}{n}}$$ is convergent.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$. For each statement, state if it is true or false.
(a) There is no continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx < \frac{1}{4}$.
(b) There is only one continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.
(c) There are infinitely many continuous functions $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.

(A) (6 marks) Let $f , g : [ 0,1 ] \mapsto \mathbb { R }$ be monotonically increasing continuous functions. Show that
$$\left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right) \leq \int _ { 0 } ^ { 1 } f ( x ) g ( x ) d x$$
(Hint: try double integrals.)
(B) (4 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be an infinitely differentiable function such that $f ( 1 ) = f ( 0 ) = 0$. Also, suppose that for some $n > 0$, the first $n$ derivatives of $f$ vanish at zero. Then prove that for the $( n + 1 )$ th derivative of $f$, $f ^ { ( n + 1 ) } ( x ) = 0$ for some $x \in ( 0,1 )$.

Let $f$ be a continuous real-valued function on $[ 0,1 ]$ such that
$$\int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } x f ( x ) d x = 0$$
Pick the correct statement(s) from below.
(A) $f$ must have a zero in $[ 0,1 ]$.
(B) $f$ has at least two zeros, counted with multiplicity, in $[ 0,1 ]$.
(C) If $f \not\equiv 0$, then $f$ has exactly two zeros in $[ 0,1 ]$.
(D) $f \equiv 0$.

Let $f$ be a real function, defined, continuous and decreasing on $[ a , + \infty [$, where $a \in \mathbb { R }$. Show that for every integer $k \in \left[ a + 1 , + \infty \left[ \right. \right.$, we have $\int _ { k } ^ { k + 1 } f ( x ) d x \leqslant f ( k ) \leqslant \int _ { k - 1 } ^ { k } f ( x ) d x$.

Let $\mathcal{A}$ and $\mathcal{B}$ be two open bounded non-empty subsets of $\mathbb{R}^{2}$ and $\lambda \in ]0,1[$. Verify that $\lambda\mathcal{A} + (1-\lambda)\mathcal{B}$ is an open bounded subset of $\mathbb{R}^{2}$. Then show that $$V(\lambda\mathcal{A} + (1-\lambda)\mathcal{B}) \geq V(\mathcal{A})^{\lambda} V(\mathcal{B})^{1-\lambda}$$ To prove this inequality, you will use the following admitted result. For all $f \in C(\mathcal{A})$ and $g \in C(\mathcal{B})$, the function $h$ determined by: $$\forall Z \in \mathbb{R}^{2},\quad h(Z) = \sup\left\{f(X)^{\lambda} g(Y)^{1-\lambda} \,/\, X, Y \in \mathbb{R}^{2},\, Z = \lambda X + (1-\lambda) Y\right\}$$ defines a continuous function on $\mathbb{R}^{2}$.

Let $u : \mathbb{R}^{2} \rightarrow ]0,+\infty[$ be a continuous and log-concave function in the sense of Part II. Prove that the preceding inequality remains true if we replace the application $V$ by the application $\gamma$ defined for all open bounded (non-empty) subsets $\mathcal{A}$ of $\mathbb{R}^{2}$ by $$\gamma(\mathcal{A}) = \sup_{f \in C(\mathcal{A})} \iint_{\mathbb{R}^{2}} f(x,y)\,u(x,y)\,dx\,dy$$

For all $i \in \{1,2,3,4\}$, we set $$\theta(N_{i}) = \frac{1}{2N_{i}} + \int_{0}^{+\infty} \frac{h(u)}{(u+N_{i})^{2}} du$$
VI.C.1) Show that for all $i \in \{1,2,3,4\}$ $$0 < \theta(N_{i}) < \frac{1}{N_{i}}$$
VI.C.2) Show the existence of a strictly positive real $K$ such that for all $i \in \{1,2,3,4\}$ $$N_{i} = K e^{\mu \varepsilon_{i}} e^{-\theta(N_{i})}$$

Show that, as $x$ tends to $+ \infty$, $$2 \pi \ln ( x ) - \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$$ has a limit, which one will determine.

Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Show that $\lim_{n \rightarrow +\infty} I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \mathrm{d}\xi$.

Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Calculate $\lim_{n \rightarrow +\infty} J_{n}$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $\delta \in ]0, \pi[$ and all real $\varphi$, $\int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z)\, \mathrm{d}t \xrightarrow[z \to \mathrm{e}^{\mathrm{i}\varphi}]{} 0$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Using Heine's theorem, show that, for all $\varepsilon > 0$, there exists $\delta > 0$ such that, for all real number $\varphi$ and all complex number $z$ satisfying $|z| < 1$, $$|g(z) - h(\varphi)| \leqslant \frac{\sup_{t \in \mathbb{R}} |h(t)|}{\pi} \int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z)\, \mathrm{d}t + \varepsilon$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $\delta \in ]0, \pi[$ and all real $\varphi$, $$\int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t \xrightarrow[z \rightarrow \mathrm{e}^{\mathrm{i}\varphi}]{} 0$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Using Heine's theorem, show that, for all $\varepsilon > 0$, there exists $\delta > 0$ such that, for all real number $\varphi$ and all complex number $z$ satisfying $|z| < 1$, $$|g(z) - h(\varphi)| \leqslant \frac{\sup_{t \in \mathbb{R}} |h(t)|}{\pi} \int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t + \varepsilon$$

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. Show that we define an inner product on $E_1$ by setting $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t)\,\mathrm{d}t$$

Show that, for every function $f:[0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have $$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2\,\mathrm{d}t}$$

In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. Show that we define an inner product on $E_1$ by setting $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$$

In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. Show that, for every function $f : [0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have $$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2 \, \mathrm{d}t}$$

Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that, for all functions $f$ and $g$ in $E$, the product $fgw$ is integrable on $I$. You may use the inequality $\forall (a,b) \in \mathbb{R}^2, |ab| \leqslant \frac{1}{2}(a^2 + b^2)$, after justifying it.