Prove there exists $f \in C^{\infty}$, bounded derivatives,

Question

I know the title isn't very comprehensible, but I don't know how to improve it.

Here is a problem which I don't know how to solve.

Let $n \in \mathbb{N}, \ \ \alpha \in \mathbb{R}, \ \ \varepsilon > 0$. Prove that there exists $f$ class $C ^{\infty}$ such that

$|f^{(k)}(x)|\le \varepsilon$ for $k = 0,1,...,n-1$, $ \ \ x \in \mathbb{R}$,

$f^{(k)}(0)=0$ for $k=0,1,...,n-1$

$f^{(n)}(0)=\alpha$.

Could you help me with that?

Answer 1

Consider $$f(x) = c(\varepsilon,n,\alpha,\beta) x^n e^{-\beta x^2}$$ Choose $c(\varepsilon,n,\alpha,\beta)$ to ensure $|f^{(k)}(x)|\le \varepsilon$ and $f^{(n)}(0)=\alpha$.