(Solved): (a) State the Inverse Function Theorem. Be sure to include all hypotheses. (b) Let \( \alpha:[a, b ...

(a) State the Inverse Function Theorem. Be sure to include all hypotheses. (b) Let \( \alpha:[a, b] \rightarrow \mathbb{R}^{n} \) be a regular \( C^{k} \) curve. Prove that the are length function \( s:[a, b] \rightarrow[0, L](L=s(b)) \) defined by \( s(l)=\int_{a}^{t}\left\|\alpha^{\prime}(u)\right\| d u \) has a \( C^{k} \) inverse. Hint: Use the Inverse Function Theorem.