Let $\varphi_{0}$ be the function defined on $\mathbb{R}$ by
$$\left\{ \begin{array}{l}
\varphi_{0}(x) = e^{-1/x^{2}} \text{ if } x \neq 0 \\
\varphi_{0}(0) = 0
\end{array} \right.$$
Using $\varphi_{0}$, show that there exists a function $\varphi_{1}$ of class $C^{\infty}$ on $\mathbb{R}$, whose support is $[0, \infty[$. Deduce that there exists a function $\varphi_{2}$ of class $C^{\infty}$ on $\mathbb{R}$ such that $\operatorname{Supp}(\varphi_{2}) = [-1, 1]$.