State and prove the Cauchy Mean Value Theorem.

Let $\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{x}^{3}$. Find a suitable \((\mathrm{u}, \mathrm{v})\) on the line segment connecting \((\mathrm{a}, b)\) with \((c, d)\) such that $\mathrm{f}(\mathrm{c}, \mathrm{d})-\mathrm{f}(\mathrm{a}, b)=(\partial \mathrm{f} / \partial \mathrm{x})(\mathrm{u}, \mathrm{v})(\mathrm{c}-\mathrm{a})+(\partial \mathrm{f} / \partial \mathrm{y})(\mathrm{u}, \mathrm{v})(\mathrm{d}-\mathrm{b})$ if \((a, b)=(1,2)\) and \((c, d)=(1+h, 2+k)\).

Show that if a function $\mathrm{f}: \mathrm{V} \rightarrow \mathrm{R}, \mathrm{V} \subseteq \mathrm{R}^{\mathrm{n}}\(, is \)\mathrm{C}^{2}$ locally at \(\mathrm{a}\), then $\left[\left(\partial^{2} \mathrm{f}\right) /\left(\partial \mathrm{x}_{\mathrm{i}} \partial \mathrm{x}_{\mathrm{j}}\right)\right](\mathrm{a})=\left[\left(\partial^{2} \mathrm{f}\right) /\left(\partial \mathrm{x}_{j} \partial \mathrm{x}_{\mathrm{i}}\right)\right]$ (a) for all \(i, j\) between 1 and \(n\) inclusive.

Prove Taylor's Theorem for $\mathrm{f} \in \mathrm{C}^{\mathrm{T}}(\mathrm{E})\( where \)\mathrm{E} \subseteq \mathrm{R}^{\mathrm{n}}$ is an open convex set.

Represent the contour line \(y-x e^{y}=1\) near \((-1 / 0)\) as a function \(\mathrm{x}=\psi(\mathrm{y})\). Compute \(\Phi^{\prime}(\mathrm{x})\) and \(\Phi^{\prime}(-1)\) for \(\mathrm{x}\) near \(-1\), where \(\mathrm{y}=\Phi(\mathrm{x})\).

(a) Let \(\mathrm{f}: \mathrm{R}^{2} \rightarrow \mathrm{R}\) be defined by $f(x, y)=2 x y\left\\{\left(x^{2}-y^{2}\right) /\left(x^{2}+y^{2}\right)\right\\}, x^{2}+y^{2} \neq 0$ and \(=0, \quad \mathrm{x}=\mathrm{y}=0\). Show that $\left(\partial^{2} \mathbf{f} / \partial \mathrm{x} \partial \mathrm{y}\right) \neq\left(\partial^{2} \mathrm{f} / \partial \mathrm{x} \partial \mathrm{y}\right)$ and explain why. (b) Does there exist a function \(\mathrm{f}\) with continuous second partial derivatives (i.e., an element of \(\mathrm{C}^{2}\) ) such that of $/ \partial \mathrm{x}=\mathrm{x}^{2}$ and \partialf \(/ \partial \mathrm{y}=\mathrm{xy}\) ?