Let $f$ be the function defined on $\mathbb { R }$ by
$$f ( x ) = x \mathrm { e } ^ { - 2 x } .$$
We admit that $f$ is twice differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ the derivative of the function $f$. We denote $C _ { f }$ the representative curve of $f$ in an orthonormal coordinate system of the plane.
For each of the following statements, specify whether it is true or false, then justify the answer given.
Any answer without justification will not be taken into account.
Statement 1. For all real $x$, we have $f ^ { \prime } ( x ) = ( - 2 x + 1 ) \mathrm { e } ^ { - 2 x }$.
Statement 2. The function $f$ is a solution on $\mathbb { R }$ of the differential equation:
$$y ^ { \prime } + 2 y = \mathrm { e } ^ { - 2 x }$$
Statement 3. The function $f$ is convex on $] - \infty ; 1 ]$.
Statement 4. The equation $f ( x ) = - 1$ admits a unique solution on $\mathbb { R }$.
Statement 5. The area of the region bounded by the curve $C _ { f }$, the $x$-axis and the lines with equations $x = 0$ and $x = 1$ is equal to $\frac { 1 } { 4 } - \frac { 3 \mathrm { e } ^ { - 2 } } { 4 }$.