91Ó°ÊÓ

Suppose that the functions \(\psi: \mathbb{R}^{3} \rightarrow \mathbb{R}\) and \(\phi: \mathbb{R}^{3} \rightarrow \mathbb{R}\) are continuously differentiable. Define, for \((x, y, z)\) in \(\mathbb{R}^{3}\), $$ \mathbf{F}(x, y, z)=\left(\psi(x, y, z), \varphi(x, y, z),(\psi(x, y, z))^{2}+(\varphi(x, y, z))^{2}\right) $$ a. Explain analytically why there is no point \(\left(x_{0}, y_{0}, z_{0}\right)\) in \(\mathbb{R}^{3}\) at which the assumptions of the Inverse Function Theorem hold for the mapping \(\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\). b. Explain geometrically why there is no point \(\left(x_{0}, y_{0}, z_{0}\right)\) in \(\mathbb{R}^{3}\) at which the conclusion of the Inverse Function Theorem holds for the mapping \(\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\).

a. Give an example of a continuously differentiable function \(f: \mathbb{R} \rightarrow \mathbb{R}\) that is one-to-one but not onto. b. Provide an example of a continuously differentiable function \(f: \mathbb{R} \rightarrow \mathbb{R}\) that is onto but not one-to-one.

Define \(f(x)=x^{3}\) for \(x\) in \(\mathbb{R}\). Show that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is one-to-one and onto and that its inverse \(f^{-1}: \mathbb{R} \rightarrow \mathbb{R}\) is continuous. At what points is the inverse differentiable?

Let \(U\) be an open subset of \(\mathbb{R}^{n}\) and suppose that the continuously differentiable mapping \(\mathbf{F}: U \rightarrow \mathbb{R}^{n}\) is stable. Prove that at each point \(\mathbf{x}\) in \(U,\) the derivative matrix DF \((x)\) is invertible. (Hint: Use the First-Order Approximation Theorem.)