Let $c:[-1,1]\to\mathbb{R}^2$ a $\mathcal{C}^1$ planar curve and suppose that $c(0)=(0,0)$ and $c'(0,0)=(a,0)$, $a>0$. I'm trying to prove the following statement (without any success): there exist $m,q>0$ such that the curve $c$ is contained in the triangle of vertices $(0,0),(q,mq),(q,-mq)$ for every $t$ in a sufficiently small right neighborhood of $0$.
Intuitively it has to be true since tangent vectors approximate the curve locally, but how to prove it rigorously?
Thank you in advance.