# 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.