We consider $f$ the function defined on the interval $] 0 ; + \infty \left[ \right.$ by $f ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } } { 2 \sqrt { x } }$ and we call $C _ { f }$ its representative curve in an orthonormal coordinate system.

\begin{enumerate}
  \item We define the function $g$ on the interval $] 0 ; + \infty \left[ \operatorname { by } g ( x ) = \mathrm { e } ^ { \sqrt { x } } \right.$.\\(a) Show that $g ^ { \prime } ( x ) = f ( x )$ for all $x$ in the interval $] 0 ; + \infty [$.\\(b) For all real $x$ in the interval $] 0 ; + \infty \left[ \right.$, calculate $f ^ { \prime } ( x )$ and show that:
\end{enumerate}

$$f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( \sqrt { x } - 1 ) } { 4 x \sqrt { x } }$$

\begin{enumerate}
  \setcounter{enumi}{1}
  \item (a) Determine the limit of the function $f$ at 0.\\(b) Interpret this result graphically.
  \item (a) Determine the limit of the function $f$ at $+ \infty$.\\(b) Study the direction of variation of the function $f$ on $] 0 ; + \infty [$.\\Draw the variation table of the function $f$ including the limits at the boundaries of the domain of definition.\\(c) Prove that the equation $f ( x ) = 2$ has a unique solution on the interval [ $1 ; + \infty [$ and give an approximate value to $10 ^ { - 1 }$ of this solution.
  \item We set $I = \int _ { 1 } ^ { 2 } f ( x ) d x$.\\(a) Calculate $I$.\\(b) Interpret graphically the result obtained.
  \item We admit that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$ and that:

$$f ^ { \prime \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( x - 3 \sqrt { x } + 3 ) } { 8 x ^ { 2 } \sqrt { x } }$$

(a) By setting $X = \sqrt { x }$, show that $x - 3 \sqrt { x } + 3 > 0$ for all real $x$ in the interval $] 0 ; + \infty [$.\\(b) Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$.
\end{enumerate}