Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by

$$f ( x ) = x ^ { 2 } - 6 x + 4 \ln ( x )$$

It is admitted that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$.
We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative.
We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthogonal coordinate system.

\begin{enumerate}
  \item a. Determine $\lim _ { x \rightarrow 0 } f ( x )$.

Interpret this result graphically.\\
b. Determine $\lim _ { x \rightarrow + \infty } f ( x )$.
  \item a. Determine $f ^ { \prime } ( x )$ for all real $x$ belonging to $] 0 ; + \infty [$.\\
b. Study the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$.

Deduce the variation table of $f$.
  \item Show that the equation $f ( x ) = 0$ has a unique solution in the interval $[4;5]$.
  \item It is admitted that, for all $x$ in $] 0 ; + \infty [$, we have:

$$f ^ { \prime \prime } ( x ) = \frac { 2 x ^ { 2 } - 4 } { x ^ { 2 } }$$

a. Study the convexity of the function $f$ on $] 0 ; + \infty [$.

The exact coordinates of any inflection points of $\mathscr { C } _ { f }$ will be specified.\\
b. We denote A the point with coordinates $( \sqrt { 2 } ; f ( \sqrt { 2 } ) )$.

Let $t$ be a strictly positive real number such that $t \neq \sqrt { 2 }$. Let $M$ be the point with coordinates $( t ; f ( t ) )$.\\
Using question 4. a, indicate, according to the value of $t$, the relative positions of the segment [AM] and the curve $\mathscr { C } _ { f }$.
\end{enumerate}