Q:

Little exercise on smooth funcions

"Let $f\in C^\infty(\mathbb{R}^N)$, then there exists $f_1,f_2\in C^\infty(\mathbb{R}^N)$ positive such that for any $x\in\mathbb{R}^N$, $f(x)=f_1(x)-f_2(x)$."

A:

Try considering
$$f_1(x) = f(x)^2 + f(x) + 1$$
And
$$f_2(x) = f(x)^2 + 1$$

