Suppose $\displaystyle f$ is infinitely differentiable on $\displaystyle \mathbb{R}$.

(a) If $\displaystyle \left|{f(x)\right|\leq 1 + \left|x\right|^m $ for some $\displaystyle m\in \mathbb {N},$ and for all $\displaystyle x\in\mathbb{R},$then $\displaystyle f(x)$ is a polynomial.