Proving function is $C^k$

Here's a start: Suppose $f \in C^2(\mathbb {R}).$ Note $f$ even implies $f'$ is odd. This is key, because it tells us$f'(0)=0.$ So letting $P_2$ denote the second degree Taylor polynomial of $f$ at $0,$ we have $P_2(t) = f(0) + f''(0)t^2/2.$ Clearly $P_2(|x|)\in C^\infty(\mathbb {R}^n).$

Set $g(t)= f(t)-P_2(t).$ Then $g\in C^2$ and $0=g(0)=g'(0)=g''(0).$ So it's enough to check that $G(x) = g(|x|)$ is $C^2.$ We certainly have $G \in C^2(\mathbb {R}^n\setminus \{0\}).$ Suppose $D$ is a partial derivative of order $\le 2.$ If we show that $D(G)$ tends to $0$ at the origin, then it will follow that $G \in C^2(\mathbb {R}^n).$