Let $f$ and $g$ be the functions defined on $\mathbb{R}$ by
$$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 2\mathrm{e}^{\frac{x}{2}} - 1.$$
We denote $\mathcal{C}_f$ and $\mathcal{C}_g$ the representative curves of functions $f$ and $g$ in an orthogonal coordinate system.

\begin{enumerate}
  \item Prove that the curves $\mathcal{C}_f$ and $\mathcal{C}_g$ have a common point with abscissa 0 and that at this point, they have the same tangent line $\Delta$ whose equation we will determine.
  \item Study of the relative position of curve $\mathcal{C}_g$ and line $\Delta$

Let $h$ be the function defined on $\mathbb{R}$ by $h(x) = 2\mathrm{e}^{\frac{x}{2}} - x - 2$.\\
a. Determine the limit of function $h$ at $-\infty$.\\
b. Justify that, for every real $x \neq 0, h(x) = x\left(\frac{\mathrm{e}^{\frac{x}{2}}}{\frac{x}{2}} - 1 - \frac{2}{x}\right)$.\\
Deduce the limit of function $h$ at $+\infty$.\\
c. We denote $h'$ the derivative function of function $h$ on $\mathbb{R}$.\\
For every real $x$, calculate $h'(x)$ and study the sign of $h'(x)$ according to the values of $x$.\\
d. Draw the variation table of function $h$ on $\mathbb{R}$.\\
e. Deduce that, for every real $x, 2\mathrm{e}^{\frac{x}{2}} - 1 \geqslant x + 1$.\\
f. What can we deduce about the relative position of curve $\mathcal{C}_g$ and line $\Delta$?\\
  \item Study of the relative position of curves $\mathcal{C}_f$ and $\mathcal{C}_g$.\\
a. For every real $x$, expand the expression $\left(\mathrm{e}^{\frac{x}{2}} - 1\right)^2$.\\
b. Determine the relative position of curves $\mathcal{C}_f$ and $\mathcal{C}_g$.
\end{enumerate}