differential geometry – Second derivatives and three points on a straight line

Consider a smooth plane curve $c:Ito mathbb{R}^2$ with $I$ an open interval in the real line and $c’ neq 0$ everywhere. Assume $t_0 in I$. My question is whether following statements are equivalent:

(1) $c”(t_0) neq 0$

(2) There is a sequence $(h_n)_{n in mathbb{N}}$ with $h_n to 0$ and all $h_n>0$ such that for each $n$, the three points $c(t_0 pm h_n)$, $c(t_0)$ are not located on a straight line.

If this is true, what is a proof? If it is false, what is a counterexample? And if it is false, is there any modifiaction of statement (1) to make the equivalence true? Or a modification of statement (2) or both?

So far, I only tried the direction “(1) $Rightarrow$ (2)” by considering the Taylor-Formula for $c$. Didn’t get anything useful from that.