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 $$ \varphi(t) = \begin{cases} t^4,& \text{if } t\le 1,\\ 2, &\text{otherwise.} \end{cases} $$
Arnak
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