Let $\displaystyle c:[a,b] \to \mathbb{R}^2 $ be a regular plane curve such that $\displaystyle ||c(t)|| \leq 1 $ for all $\displaystyle t \in [a,b] $. Suppose there is a point $\displaystyle t_0 \in [a,b] $ such that $\displaystyle ||c(t_0)|| = 1 $. Prove that the curvature at that point satisfies $\displaystyle |\kappa (t_0)| \geq 1 $.