Exercise 2, § 3

Let $x_0\in \mathbf{R}^n,r >0$ and $t >1$. There exists $\varphi \in \mathscr{D}(\mathbf{R}^n)$ such that

(i) $0\leqslant \varphi \leqslant 1$;

(ii) $\varphi (x) = 1$ for $\|x-x_0\|\leqslant r$ and $\varphi (x) = 0$ for $\|x-x_0\|\geqslant tr$;

(iii) For every $\alpha \in \mathbf{N}^n$, one has

sup$_{x\in\mathbf{R}^n}|\partial^{\alpha}\varphi (x)|\leqslant \alpha$!(($t-1$)$r$)$^{|\alpha|}$.