Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x - \ln\left(x^{2} + 1\right).$$
Solve in $\mathbb{R}$ the equation: $f(x) = x$.
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
+
$+\infty$
$f(x)$
$-\infty$
Show that, for every real $x$ belonging to $[0;1]$, $f(x)$ belongs to $[0;1]$.
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 $f$ be the function defined on $\mathbb{R}$ by
$$f(x) = x - \ln\left(x^{2} + 1\right).$$
\begin{enumerate}
\item Solve in $\mathbb{R}$ the equation: $f(x) = x$.
\item Justify all elements of the variation table below except for the limit of the function $f$ at $+\infty$ which is admitted.
\begin{center}
\begin{tabular}{ | c | c c c c c c | }
\hline
$x$ & $-\infty$ & & 1 & & $+\infty$ & \\
\hline
$f^{\prime}(x)$ & & + & 0 & + & & \\
\hline
& & & & & & $+\infty$ \\
$f(x)$ & & & & & & \\
& $-\infty$ & & & & & \\
\hline
\end{tabular}
\end{center}
\item Show that, for every real $x$ belonging to $[0;1]$, $f(x)$ belongs to $[0;1]$.
\item Consider the following algorithm:
\begin{center}
\begin{tabular}{ | l | l | }
\hline
Variables & $N$ and $A$ natural integers; \\
\hline
Input & Enter the value of $A$ \\
\hline
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 \\
\end{tabular} \\
\hline
Output & Display $N$ \\
\hline
\end{tabular}
\end{center}
a. What does this algorithm do?\\
b. Determine the value $N$ provided by the algorithm when the value entered for $A$ is 100.
\end{enumerate}