For each real number $a$, we consider the function $f _ { a }$ defined on the set of real numbers $\mathbb { R }$ by

$$f _ { a } ( x ) = \mathrm { e } ^ { x - a } - 2 x + \mathrm { e } ^ { a } .$$

\begin{enumerate}
  \item Show that for every real number $a$, the function $f _ { a }$ has a minimum.
  \item Does there exist a value of $a$ for which this minimum is as small as possible?
\end{enumerate}