Let $N>4$ and $2a>N$. Consider the function $f:\mathbb R^N\to\mathbb R$ be a function of class $C^2$ such that $$|f(x)|\le \frac{1}{(1+|x|^2)^{\alpha/2}}.$$
Is it possible to use this information to deduce that there exists a constant $c>0$ such that $$\|\nabla f\|_{\infty} \le c?$$
At calc class today, it was exhibited as an obvious thing while proving a theorem.
Could someone please help me understanding why is that true? If it is not, which extra assumption on $f$ could guarantee that $\|\nabla f\|_{\infty}\le c?$
I am a 2 year physics student and I am not so familiar with this kind of things. Thank you everyone.