Does there exist a $C_c^\infty(\mathbb R)$ function $f$ such that $\sup_x |x|^k|f^{(k)}(x)|\leq M^k$ for all $k$ and fixed $M$? [closed]

I know that for a $C_c^\infty(\mathbb R)$ function $f$, and for each given $k\in\mathbb N$, we can give existence of $M_k$ such that

$$\sup_x |x|^k|f^{(k)}(x)|\leq M_k^k$$

as $f$ is a Schwartz function. But my expectation is something stronger. I want to understand whether we can find any $f$ for which $M_k$ can be chosen uniformly on $k$.