Extend function with absolute value smoothly

Question

I am dealing with the function $$f(x)=C_{1}-C_{2}|x|^{-\alpha}$$ in $\mathbb{R}^{n}\setminus B_{1/4}$, where $\alpha \geq 1$. How can I extend $f$ to all $\mathbb{R}^{n}$ (and this extension depends only on $\alpha$) or see that this extension at least exists? In addition $f$ has to satisfy $f\leq-2$ in $Q_{3/2}=\{x\in \mathbb{R}^{n}:|x|_{\infty}<3/2\}$.

Answer 1

Hint: use a bump function: you can easily construct a $C^\infty$ function $h$ with $h(x) = 1$ for $x\in\mathbb{R}^{n}\setminus B_{1/4}$ and $h(x) = 0$ for $x\in B_{1/8}$.

Simply $fh$ will not work because you want $fh\le -2$ in the cube. Solution: $$(f + 2)h - 2.$$