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

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.