We consider the function $f$ defined on $] 0 ; + \infty [$ by:

$$f ( x ) = 3 x + 1 - 2 x \ln ( x ) .$$

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

\begin{enumerate}
  \item Determine the limit of the function $f$ at 0 and at $+ \infty$.
  \item a. Prove that for every strictly positive real number $x$: $f ^ { \prime } ( x ) = 1 - 2 \ln ( x )$.\\
b. Study the sign of $f ^ { \prime }$ and draw up the variation table of the function $f$ on the interval $] 0 ; + \infty [$. This table should include the limits as well as the exact value of the extremum.
  \item a. Prove that the equation $f ( x ) = 0$ has a unique solution on $] 0 ; + \infty [$. We denote this solution by $\alpha$.\\
b. Deduce the sign of the function $f$ on $] 0 ; + \infty [$.
  \item We consider any primitive of the function $f$ on the interval $] 0$; $+ \infty [$. We denote it by $F$. Can we assert that the function $F$ is strictly decreasing on the interval $\left[ \mathrm { e } ^ { \frac { 1 } { 2 } } ; + \infty [ \right.$ ? Justify.
  \item a. Study the convexity of the function $f$ on $] 0 ; + \infty [$. What is the position of the curve $\mathscr { C } _ { f }$ relative to its tangent lines?\\
b. Determine an equation of the tangent line $T$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 1.\\
c. Deduce from questions 5.a and 5.b that for every strictly positive real number $x$:
$$\ln ( x ) \geqslant 1 - \frac { 1 } { x } .$$
\end{enumerate}