I am trying to self teach differential geometry and to that effect I am trying to do the Homework in the MIT open course.
The specific question I am struggling with is:
Let $c$ be a regular curve such that $|c(s)| ≤ 1$ for all $s$. Suppose
that there is a point $t$ where $|c(t)| = 1$. Prove that the curvature at that
point satisfies $|κ(t)| ≥ 1$
This is what I have so far: The curve being regular implies that it can be arc length parametrized, which means that it's curvature is just the norm of the second derivative, or in 2D $|x'y'' - x''y'|$.
The curve magnitude being upper bounded by 1 means the curve is fully contained within the unit disk.
The curve being one at $t$ means that point is ON the unit circle. Thus my intuition says this curve can only "bend inwards" at a rate equal or higher than the circle, otherwise it will be an epsilon outside of the circle at a point infinitesimally close to $t$. In other words it seems to me that a if a curve with a point on the unit circle has a smaller curvature than that of the unit circle (i.e 1) the curve will "pop out" of it for some infinitesimal amount. But I am not sure if A) this is correct B) how to formalize it.
I am looking mostly for a hint and advice, not for the full solution. Thank you lots in advance.
Let $t_0$ be the point satisfying $\| \alpha(t_0)\| = 1$. Then, at $t_0$, the function $$t \mapsto \|\alpha(t) \|^2$$
assumes a maximum. Thus, its first order derivative at $t_0$ vanishes, that is:
$$2 \langle \alpha(t_0), \alpha'(t_0)\rangle = 0$$
And by the second derivative test, its second order derivative at $t_0$ is at most $0$, that is:
$$2 \langle \alpha'(t_0), \alpha'(t_0)\rangle + 2 \langle \alpha''(t_0), \alpha'(t_0) \rangle \leq 0$$
Assuming without loss of generality that $\alpha$ has unit speed, we get:
$$1 \leq \langle \alpha''(t_0), \alpha'(t_0) \rangle $$
Taking the norm of both sides and using the Cauchy-Schwarz inequality, we have:
$$1 \leq | \langle \alpha''(t_0), \alpha'(t_0) \rangle | \leq \|\alpha''(t_0)\| \cdot \|\alpha'(t_0)\| = |\kappa(t_0)|$$
as desired.