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.


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

  • $\begingroup$ I have considered that before, but after differentiate and equal to $0$ the result at $t_0$ what can I say? $\endgroup$ Oct 31 '14 at 3:26
  • $\begingroup$ More calculus knowledge: You know more about $t_0$ than just that it's a critical point. $\endgroup$ Oct 31 '14 at 3:32
  • $\begingroup$ Sorry but I can't see your point... Can you be a little more specific? $\endgroup$ Oct 31 '14 at 3:51
  • $\begingroup$ Could you please give me the solution or another hint? I need to solve this problem $\endgroup$ Oct 31 '14 at 13:07
  • $\begingroup$ That's the point: You need to solve it. What did the problem tell you happens at $t_0$? How did you use that? You know more than $f’=0$ at that point. $\endgroup$ Oct 31 '14 at 14:43

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