The claim is false. The counter example is the function 
$f(x) = \varphi(\|x\|_2)$ with an increasing and infinitely differentiable
function $\varphi:[0,\infty)\to\mathbb R$ satisfying
$$
\varphi(t) = 
\begin{cases}
t^4,& \text{if } t\le 1,\\
2, &\text{otherwise.}
\end{cases}
$$