We consider the function $g$ defined for all real $x$ in the interval $[0;1]$ by: $$g(x) = 1 + \mathrm{e}^{-x}$$ We admit that, for all real $x$ in the interval $[0;1], g(x) > 0$.
We denote $\mathscr{C}$ the representative curve of function $g$ in an orthogonal coordinate system, and $\mathscr{D}$ the plane region bounded on one hand between the $x$-axis and curve $\mathscr{C}$, on the other hand between the lines with equations $x = 0$ and $x = 1$.
The purpose of this exercise is to divide region $\mathscr{D}$ into two regions of equal area, first by a line parallel to the $y$-axis (part A), then by a line parallel to the $x$-axis (part B).

Part A

Let $a$ be a real number such that $0 \leqslant a \leqslant 1$. We denote $\mathscr{A}_1$ the area of the region between curve $\mathscr{C}$, the $x$-axis, the lines with equations $x = 0$ and $x = a$, and $\mathscr{A}_2$ that of the region between curve $\mathscr{C}$, the $x$-axis and the lines with equations $x = a$ and $x = 1$. $\mathscr{A}_1$ and $\mathscr{A}_2$ are expressed in square units.

1. a. Prove that $\mathscr{A}_1 = a - \mathrm{e}^{-a} + 1$. b. Express $\mathscr{A}_2$ as a function of $a$.
2. Let $f$ be the function defined for all real $x$ in the interval $[0;1]$ by: $$f(x) = 2x - 2\mathrm{e}^{-x} + \frac{1}{\mathrm{e}}$$ a. Draw the variation table of function $f$ on the interval $[0;1]$. The exact values of $f(0)$ and $f(1)$ will be specified. b. Prove that function $f$ vanishes once and only once on the interval $[0;1]$, at a real number $\alpha$. Give the value of $\alpha$ rounded to the nearest hundredth.
3. Using the previous questions, determine an approximate value of the real $a$ for which the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal.

Part B

Let $b$ be a positive real number. In this part, we propose to divide region $\mathscr{D}$ into two regions of equal area by the line with equation $y = b$. We admit that there exists a unique positive real $b$ that is a solution.

1. Justify the inequality $b < 1 + \frac{1}{\mathrm{e}}$. You may use a graphical argument.
2. Determine the exact value of the real $b$.

A factory produces mineral water in bottles. The shape of the bottle labels is bounded by the x-axis and the curve $\mathscr { C }$ with equation $y = a \cos x$ with $x \in \left[ - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \right]$ and $a$ a strictly positive real number.
A disk located inside is intended to receive information given to buyers. We consider the disk with centre at point A with coordinates $\left( 0 ; \frac { a } { 2 } \right)$ and radius $\frac { a } { 2 }$. It is admitted that this disk is entirely below the curve $\mathscr { C }$ for values of $a$ less than 1.4.

1. Justify that the area of the region between the x-axis, the lines with equations $x = - \frac { \pi } { 2 }$ and $x = \frac { \pi } { 2 }$, and the curve $\mathscr { C }$ equals $2 a$ square units.
2. For aesthetic reasons, it is desired that the area of the disk equals the area of the shaded surface. What value should be given to the real number $a$ to satisfy this constraint?

Let $f$ be a function defined on the interval $[0;1]$, continuous and positive on this interval, and $a$ a real number such that $0 < a < 1$.
We denote:

• $\mathscr{C}$ the representative curve of the function $f$ in an orthogonal coordinate system;
• $\mathscr{A}_1$ the area of the plane region bounded by the $x$-axis and the curve $\mathscr{C}$ on one hand, the lines with equations $x = 0$ and $x = a$ on the other hand.
• $\mathscr{A}_2$ the area of the plane region bounded by the $x$-axis and the curve $\mathscr{C}$ on one hand, the lines with equations $x = a$ and $x = 1$ on the other hand.

The purpose of this exercise is to determine, for different functions $f$, a value of the real number $a$ satisfying condition (E): ``the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal''. We admit the existence of such a real number $a$ for each of the functions considered.
Part A: Study of some examples

1. Verify that in the following cases, condition (E) is satisfied for a unique real number $a$ and determine its value. a. $f$ is a strictly positive constant function. b. $f$ is defined on $[0;1]$ by $f(x) = x$.
   
2. a. Using integrals, express, in units of area, the areas $\mathscr{A}_1$ and $\mathscr{A}_2$. b. Let $F$ be a primitive of the function $f$ on the interval $[0;1]$. Prove that if the real number $a$ satisfies condition (E), then $F(a) = \dfrac{F(0) + F(1)}{2}$. Is the converse true?
   
3. In this question, we consider two other particular functions. a. The function $f$ is defined for all real $x$ in $[0;1]$ by $f(x) = \mathrm{e}^x$. Verify that condition (E) is satisfied for a unique real number $a$ and give its value. b. The function $f$ defined for all real $x$ in $[0;1]$ by $f(x) = \dfrac{1}{(x+2)^2}$. Verify that the value $a = \dfrac{2}{5}$ works.

Part B: Using a sequence to determine an approximate value of $a$
In this part, we consider the function $f$ defined for all real $x$ in $[0;1]$ by $f(x) = 4 - 3x^2$.

1. Prove that if $a$ is a real number satisfying condition (E), then $a$ is a solution of the equation: $$x = \frac{x^3}{4} + \frac{3}{8}$$ In the rest of the exercise, we will admit that this equation has a unique solution in the interval $[0;1]$. We denote this solution by $a$.
   
2. We consider the function $g$ defined for all real $x$ in $[0;1]$ by $g(x) = \dfrac{x^3}{4} + \dfrac{3}{8}$ and the sequence $(u_n)$ defined by: $u_0 = 0$ and, for all natural number $n$, $u_{n+1} = g(u_n)$. a. Calculate $u_1$. b. Prove that the function $g$ is increasing on the interval $[0;1]$. c. Prove by induction that, for all natural number $n$, we have $0 \leqslant u_n \leqslant u_{n+1} \leqslant 1$. d. Prove that the sequence $(u_n)$ is convergent. Using operations on limits, prove that the limit is $a$. e. We admit that the real number $a$ satisfies the inequality $0 < a - u_{10} < 10^{-9}$. Calculate $u_{10}$ to $10^{-8}$ precision.

For all functions $f ( x )$ that are differentiable on the set of all real numbers and satisfy the following conditions, what is the minimum value of $\int _ { 0 } ^ { 2 } f ( x ) d x$? [4 points] (가) $f ( 0 ) = 1 , f ^ { \prime } ( 0 ) = 1$ (나) If $0 < a < b < 2$, then $f ^ { \prime } ( a ) \leqq f ^ { \prime } ( b )$. (다) On the interval $( 0,1 )$, $f ^ { \prime \prime } ( x ) = e ^ { x }$.
(1) $\frac { 1 } { 2 } e - 1$
(2) $\frac { 3 } { 2 } e - 1$
(3) $\frac { 5 } { 2 } e - 1$
(4) $\frac { 7 } { 2 } e - 2$
(5) $\frac { 9 } { 2 } e - 2$

Find the positive value of $a$ that satisfies $\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } - 4 \right) d x = 0$. [3 points]
(1) 2
(2) $\frac { 9 } { 4 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 11 } { 4 }$
(5) 3

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac{3}{4}$
(B) $\frac{4}{3}$
(C) $\frac{3}{2}$
(D) $\frac{2}{3}$

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac{3}{4}$
(B) $\frac{4}{3}$
(C) $\frac{3}{2}$
(D) $\frac{2}{3}$

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac { 3 } { 4 }$
(B) $\frac { 4 } { 3 }$
(C) $\frac { 3 } { 2 }$
(D) $\frac { 2 } { 3 }$

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(a) $3 / 4$.
(B) $4 / 3$.
(C) $3 / 2$.
(D) $2 / 3$.

The total number of distinct $x \in [0,1]$ for which $\int_0^x \frac{t^2}{1+t^4}\,dt = 2x - 1$ is

If $n ( 2 n + 1 ) \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n } d x = 1177 \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n + 1 } d x$, then $n \in N$ is equal to $\_\_\_\_$.

The minimum value of the function $f ( x ) = \int _ { 0 } ^ { 2 } e ^ { | x - t | } d t$ is
(1) $2 ( e - 1 )$
(2) $2 e - 1$
(3) 2
(4) $e ( e - 1 )$

If the value of the integral $\int _ { - 1 } ^ { 1 } \frac { \cos \alpha x } { 1 + 3 ^ { x } } d x$ is $\frac { 2 } { \pi }$. Then, a value of $\alpha$ is
(1) $\frac { \pi } { 3 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$