\textbf{Part A}

Let $g$ be the function defined on the set of real numbers $\mathbf { R }$, by
$$g ( x ) = x ^ { 2 } + x + \frac { 1 } { 4 } + \frac { 4 } { \left( 1 + \mathrm { e } ^ { x } \right) ^ { 2 } }$$

It is admitted that the function $g$ is differentiable on $\mathbf { R }$ and we denote by $g ^ { \prime }$ its derivative function.

1. Determine the limits of $g$ at $+ \infty$ and at $- \infty$.

2. It is admitted that the function $g ^ { \prime }$ is strictly increasing on $\mathbf { R }$ and that $g ^ { \prime } ( 0 ) = 0$.

Determine the sign of the function $g ^ { \prime }$ on $\mathbf { R }$.

3. Draw up the table of variations of the function $g$ and calculate the minimum of the function $g$ on $\mathbf { R }$.

\textbf{Part B}

Let $f$ be the function defined on $\mathbf { R }$ by:
$$f ( x ) = 3 - \frac { 2 } { 1 + \mathrm { e } ^ { x } }$$

We denote by $\mathscr { C } _ { f }$ the representative curve of $f$ in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).

Let A be the point with coordinates $\left( - \frac { 1 } { 2 } ; 3 \right)$.

1. Prove that point $\mathrm { B } ( 0 ; 2 )$ belongs to $\mathscr { C } _ { f }$.

2. Let $x$ be any real number. We denote by $M$ the point on the curve $\mathscr { C } _ { f }$ with coordinates $( x ; f ( x ) )$.

Prove that $\mathrm { A } M ^ { 2 } = g ( x )$.

3. It is admitted that the distance $\mathrm { A } M$ is minimal if and only if $\mathrm { A } M ^ { 2 }$ is minimal.

Determine the coordinates of the point on the curve $\mathscr { C } _ { f }$ such that the distance AM is minimal.

4. It is admitted that the function $f$ is differentiable on $\mathbf { R }$ and we denote by $f ^ { \prime }$ its derivative function.

a. Calculate $f ^ { \prime } ( x )$ for all real $x$.

b. Let $T$ be the tangent to the curve $\mathscr { C } _ { f }$ at point B.

Prove that the reduced equation of $T$ is $y = \frac { x } { 2 } + 2$.

5. Prove that the line $T$ is perpendicular to the line (AB).