We consider the function $f$ defined on $] 0 ; + \infty [$ by $$f ( x ) = \frac { ( \ln x ) ^ { 2 } } { x }$$ We denote $\mathscr { C }$ the representative curve of $f$ in an orthonormal coordinate system.
Determine the limit of the function $f$ at 0 and interpret the result graphically.
a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$, $$f ( x ) = 4 \left( \frac { \ln ( \sqrt { x } ) } { \sqrt { x } } \right) ^ { 2 }$$ b. Deduce that the $x$-axis is an asymptote to the representative curve of the function $f$ in the neighbourhood of $+ \infty$.
We admit that $f$ is differentiable on $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function. a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$, $$f ^ { \prime } ( x ) = \frac { \ln ( x ) ( 2 - \ln ( x ) ) } { x ^ { 2 } } .$$ b. Study the sign of $f ^ { \prime } ( x )$ according to the values of the strictly positive real number $x$. c. Calculate $f ( 1 )$ and $f \left( \mathrm { e } ^ { 2 } \right)$.
Prove that the equation $f ( x ) = 1$ admits a unique solution $\alpha$ on $] 0 ; + \infty [$ and give a bound for $\alpha$ with amplitude $10 ^ { - 2 }$.
We consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = \frac { ( \ln x ) ^ { 2 } } { x }$$
We denote $\mathscr { C }$ the representative curve of $f$ in an orthonormal coordinate system.
\begin{enumerate}
\item Determine the limit of the function $f$ at 0 and interpret the result graphically.
\item a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$,
$$f ( x ) = 4 \left( \frac { \ln ( \sqrt { x } ) } { \sqrt { x } } \right) ^ { 2 }$$
b. Deduce that the $x$-axis is an asymptote to the representative curve of the function $f$ in the neighbourhood of $+ \infty$.
\item We admit that $f$ is differentiable on $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function.\\
a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$,
$$f ^ { \prime } ( x ) = \frac { \ln ( x ) ( 2 - \ln ( x ) ) } { x ^ { 2 } } .$$
b. Study the sign of $f ^ { \prime } ( x )$ according to the values of the strictly positive real number $x$.\\
c. Calculate $f ( 1 )$ and $f \left( \mathrm { e } ^ { 2 } \right)$.
\item Prove that the equation $f ( x ) = 1$ admits a unique solution $\alpha$ on $] 0 ; + \infty [$ and give a bound for $\alpha$ with amplitude $10 ^ { - 2 }$.
\end{enumerate}