Let $f$ be the function defined for all real numbers $x$ in the interval $] 0 ; + \infty [$ by:

$$f ( x ) = \frac { \mathrm { e } ^ { 2 x } } { x }$$

The expression of the second derivative $f ^ { \prime \prime }$ of $f$ is given, defined on the interval $] 0 ; + \infty [$ by:

$$f ^ { \prime \prime } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } - 2 x + 1 \right) } { x ^ { 3 } } .$$

\begin{enumerate}
  \item The function $f ^ { \prime }$, the derivative of $f$, is defined on the interval $] 0 ; + \infty [$ by:\\
a. $f ^ { \prime } ( x ) = 2 \mathrm { e } ^ { 2 x }$\\
b. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( x - 1 ) } { x ^ { 2 } }$\\
c. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 2 x - 1 ) } { x ^ { 2 } }$\\
d. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 1 + 2 x ) } { x ^ { 2 } }$.
  \item The function $f$:\\
a. is decreasing on $] 0 ; + \infty [$\\
b. is monotonic on $] 0 ; + \infty [$\\
c. admits a minimum at $\frac { 1 } { 2 }$\\
d. admits a maximum at $\frac { 1 } { 2 }$.
  \item The function $f$ has the following limit as $x \to + \infty$:\\
a. $+ \infty$\\
b. 0\\
c. 1\\
d. $\mathrm { e } ^ { 2 x }$.
  \item The function $f$:\\
a. is concave on $] 0 ; + \infty [$\\
b. is convex on $] 0 ; + \infty [$\\
c. is concave on $] 0 ; \frac { 1 } { 2 } ]$\\
d. is represented by a curve admitting an inflection point.
\end{enumerate}