The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$ We define the functions $f$ and $g$ on $]0, 2\pi]$ by: $$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$ Show that $f$ and $g$ extend by continuity to $[0, 2\pi]$; that $f$ is then a continuous bijection from $[0, 2\pi]$ onto a set to be specified; and that $g$ attains its maximum at $\pi$.
The Carnot sphere is the set:
$$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
We define the functions $f$ and $g$ on $]0, 2\pi]$ by:
$$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$
Show that $f$ and $g$ extend by continuity to $[0, 2\pi]$; that $f$ is then a continuous bijection from $[0, 2\pi]$ onto a set to be specified; and that $g$ attains its maximum at $\pi$.