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Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function with the following properties.

  1. $\phi(x) = 1$ if $|x| \leq 1$

  2. $\phi(x) = 0$ if $|x| \geq 2$

  3. $0 \leq \phi \leq 1$

  4. $\phi$ is radial.

Let $\phi_k(x) = \phi(\frac{x}{k})$. How do I show that for each multiindex $\alpha$, there exists a constant $C_\alpha$ such that $|D^\alpha \phi_k| \leq \frac{C_\alpha}{k^{\textrm{deg }\alpha}}$ uniformly in $k$?

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