Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by: $$f ( x ) = \frac { \ln ( x ) } { x ^ { 2 } } + 1$$ We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthonormal coordinate system. It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function.
Determine the limits of the function $f$ at 0 and at $+ \infty$. Deduce the possible asymptotes to the curve $\mathscr { C } _ { f }$.
Show that, for all real $x$ in the interval $] 0 ; + \infty [$, we have: $$f ^ { \prime } ( x ) = \frac { 1 - 2 \ln ( x ) } { x ^ { 3 } }$$
Deduce the variation table of the function $f$ on the interval $] 0 ; + \infty [$.
a. Show that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $] 0 ; + \infty [$. b. Give an interval for the real number $\alpha$ with amplitude 0.01. c. Deduce the sign of the function $f$ on the interval $] 0 ; + \infty [$.
Consider the function $g$ defined on the interval $] 0 ; + \infty [$ by: $$g ( x ) = \ln ( x )$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthonormal coordinate system with origin O. We consider a strictly positive real $x$ and the point M of the curve $\mathscr{C} _ { g }$ with abscissa $x$. We denote OM the distance between points O and M. a. Express the quantity $\mathrm { OM } ^ { 2 }$ as a function of the real $x$. b. Show that, when the real $x$ ranges over the interval $] 0 ; + \infty [$, the quantity $\mathrm { OM } ^ { 2 }$ admits a minimum at $\alpha$. c. The minimum value of the distance OM, when the real $x$ ranges over the interval $] 0 ; + \infty [$, is called the distance from point O to the curve $\mathscr { C } _ { g }$. We denote $d$ this distance. Express $d$ in terms of $\alpha$.
Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by:
$$f ( x ) = \frac { \ln ( x ) } { x ^ { 2 } } + 1$$
We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthonormal coordinate system. It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function.
\begin{enumerate}
\item Determine the limits of the function $f$ at 0 and at $+ \infty$.\\
Deduce the possible asymptotes to the curve $\mathscr { C } _ { f }$.
\item Show that, for all real $x$ in the interval $] 0 ; + \infty [$, we have:
$$f ^ { \prime } ( x ) = \frac { 1 - 2 \ln ( x ) } { x ^ { 3 } }$$
\item Deduce the variation table of the function $f$ on the interval $] 0 ; + \infty [$.
\item a. Show that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $] 0 ; + \infty [$.\\
b. Give an interval for the real number $\alpha$ with amplitude 0.01.\\
c. Deduce the sign of the function $f$ on the interval $] 0 ; + \infty [$.
\item Consider the function $g$ defined on the interval $] 0 ; + \infty [$ by:
$$g ( x ) = \ln ( x )$$
We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthonormal coordinate system with origin O. We consider a strictly positive real $x$ and the point M of the curve $\mathscr{C} _ { g }$ with abscissa $x$. We denote OM the distance between points O and M.\\
a. Express the quantity $\mathrm { OM } ^ { 2 }$ as a function of the real $x$.\\
b. Show that, when the real $x$ ranges over the interval $] 0 ; + \infty [$, the quantity $\mathrm { OM } ^ { 2 }$ admits a minimum at $\alpha$.\\
c. The minimum value of the distance OM, when the real $x$ ranges over the interval $] 0 ; + \infty [$, is called the distance from point O to the curve $\mathscr { C } _ { g }$. We denote $d$ this distance.\\
Express $d$ in terms of $\alpha$.
\end{enumerate}