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
$$
\varphi(t) = 
\begin{cases}
t^4,& \text{if } t\le 1,\\
2, &\text{otherwise.}
\end{cases}
$$
There is also the example with 
$$
\varphi(t) = 
\begin{cases}
31t^2 ,& \text{if } t\le 1,\\
3t^6-20.8t^5+51t^4-56t^3+56t^2-2.2, &\text{if } t\in[1,2],\\
96t-75.8&\text{otherwise.}
\end{cases}
$$
The resulting function $f$ is then convex, since the function $\varphi$ is convex and increasing.