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
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 $f(x) = x$.
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?
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 $f(x) = 4 - 3x^2$.
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 $a$.
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.
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:
\begin{itemize}
\item $\mathscr{C}$ the representative curve of the function $f$ in an orthogonal coordinate system;
\item $\mathscr{A}_1$ 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.
\item $\mathscr{A}_2$ 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.
\end{itemize}
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.
\textbf{Part A: Study of some examples}
\begin{enumerate}
\item 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 $f(x) = x$.
\item 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?
\item 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.
\end{enumerate}
\textbf{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 $f(x) = 4 - 3x^2$.
\begin{enumerate}
\item 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 $a$.
\item 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.
\end{enumerate}