Maximum of the length of a curve and its curvature

Question

I'm trying to solve this problem but I can't find any way to do it. Some hints or helps will be very useful and I'll be very thankful. The problem says:

Let $\alpha : (a,b) \rightarrow \mathbb{R}^2$ be a regular parametrized plane curve. Assume there exists $t_0\in (a,b)$ such that the distance from the origin to the trace of $\alpha$, $\Vert\alpha(t)\Vert$, attach a maximum at $t_0$. Show that the curvature $\kappa$ of $\alpha$ at $t_0$ satisfies $$ |\kappa (t_0)|\geq \dfrac{1}{\Vert\alpha(t_0)\Vert}.$$

This problem is in the book Differential Geometry of Curves and Surfaces, Manfredo Do Carmo.

Answer 1

HINT: Assume arclength parametrization and apply your single-variable calculus knowledge to $f(s)=|\alpha(s)|^2$.