Consider the function $f$ defined on the interval $[ 1 ; + \infty [$ by $$f ( x ) = \frac { \ln x } { x }$$ where ln denotes the natural logarithm function.
Give the limit of the function $f$ at $+ \infty$.
We admit that the function $f$ is differentiable on the interval $[ 1 ; + \infty [$ and we denote by $f ^ { \prime }$ its derivative function. a. Show that, for every real number $x \geqslant 1$, $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$. b. Justify the following sign table, giving the sign of $f ^ { \prime } ( x )$ according to the values of $x$.
$x$
1
e
$+ \infty$
$f ^ { \prime } ( x )$
+
0
-
c. Draw up the complete variation table of the function $f$.
Let $k$ be a non-negative real number. a. Show that, if $0 \leqslant k \leqslant \frac { 1 } { \mathrm { e } }$, the equation $f ( x ) = k$ admits a unique solution on the interval $[1; e]$. b. If $k > \frac { 1 } { \mathrm { e } }$, does the equation $f ( x ) = k$ admit solutions on the interval $[ 1 ; + \infty [$? Justify.
Part B
Let $g$ be the function defined on $\mathbb { R }$ by: $$g ( x ) = \mathrm { e } ^ { \frac { x } { 4 } } .$$ We consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$: $u _ { n + 1 } = e ^ { \frac { u _ { n } } { 4 } }$, that is: $u _ { n + 1 } = g \left( u _ { n } \right)$.
Justify that the function $g$ is increasing on $\mathbb { R }$.
Show by induction that, for every natural integer $n$, we have: $u _ { n } \leqslant u _ { n + 1 } \leqslant \mathrm { e }$.
Deduce that the sequence $( u _ { n } )$ is convergent.
We denote by $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ is a solution of the equation: $$\mathrm { e } ^ { \frac { x } { 4 } } = x .$$
Deduce that $\ell$ is a solution of the equation $f ( x ) = \frac { 1 } { 4 }$, where $f$ is the function studied in Part A.
Give an approximate value to $10 ^ { - 2 }$ near of the limit $\ell$ of the sequence $( u _ { n } )$.
\section*{Part A}
Consider the function $f$ defined on the interval $[ 1 ; + \infty [$ by
$$f ( x ) = \frac { \ln x } { x }$$
where ln denotes the natural logarithm function.
\begin{enumerate}
\item Give the limit of the function $f$ at $+ \infty$.
\item We admit that the function $f$ is differentiable on the interval $[ 1 ; + \infty [$ and we denote by $f ^ { \prime }$ its derivative function.\\
a. Show that, for every real number $x \geqslant 1$, $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$.\\
b. Justify the following sign table, giving the sign of $f ^ { \prime } ( x )$ according to the values of $x$.
\begin{center}
\begin{tabular}{ | c | l l l l | }
\hline
$x$ & 1 & & e & $+ \infty$ \\
\hline
$f ^ { \prime } ( x )$ & & + & 0 & - \\
\hline
\end{tabular}
\end{center}
c. Draw up the complete variation table of the function $f$.
\item Let $k$ be a non-negative real number.\\
a. Show that, if $0 \leqslant k \leqslant \frac { 1 } { \mathrm { e } }$, the equation $f ( x ) = k$ admits a unique solution on the interval $[1; e]$.\\
b. If $k > \frac { 1 } { \mathrm { e } }$, does the equation $f ( x ) = k$ admit solutions on the interval $[ 1 ; + \infty [$? Justify.
\end{enumerate}
\section*{Part B}
Let $g$ be the function defined on $\mathbb { R }$ by:
$$g ( x ) = \mathrm { e } ^ { \frac { x } { 4 } } .$$
We consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$: $u _ { n + 1 } = e ^ { \frac { u _ { n } } { 4 } }$, that is: $u _ { n + 1 } = g \left( u _ { n } \right)$.
\begin{enumerate}
\item Justify that the function $g$ is increasing on $\mathbb { R }$.
\item Show by induction that, for every natural integer $n$, we have: $u _ { n } \leqslant u _ { n + 1 } \leqslant \mathrm { e }$.
\item Deduce that the sequence $( u _ { n } )$ is convergent.
\end{enumerate}
We denote by $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ is a solution of the equation:
$$\mathrm { e } ^ { \frac { x } { 4 } } = x .$$
\begin{enumerate}
\setcounter{enumi}{3}
\item Deduce that $\ell$ is a solution of the equation $f ( x ) = \frac { 1 } { 4 }$, where $f$ is the function studied in Part A.
\item Give an approximate value to $10 ^ { - 2 }$ near of the limit $\ell$ of the sequence $( u _ { n } )$.
\end{enumerate}