Let $a$ be a real number between 0 and 1. We denote by $f _ { a }$ the function defined on $\mathbf { R }$ by:

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

We denote by $I ( a )$ the integral of the function $f _ { a }$ between 0 and 1:

$$I ( a ) = \int _ { 0 } ^ { 1 } f _ { a } ( x ) \mathrm { d } x$$

\begin{enumerate}
  \item In this question, we set $a = 0$. Determine $I ( 0 )$.
  \item In this question, we set $a = 1$. We therefore study the function $f _ { 1 }$ defined on $\mathbf { R }$ by:
$$f _ { 1 } ( x ) = \mathrm { e } ^ { x } + 1$$
a. Without detailed study, sketch the graph of the function $f _ { 1 }$ on your paper in an orthogonal coordinate system and show the number $I ( 1 )$.\\
b. Calculate the exact value of $I ( 1 )$, then round to the nearest tenth.
  \item Does there exist a value of $a$ for which $I ( a )$ equals 2? If so, give an interval of width $10 ^ { - 2 }$ containing this value.
\end{enumerate}