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Exercise $1$. Given a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$, the first-order partial derivatives are provided:$\newline$$$\frac{\partial f}{\partial x}(x, y)=6 x-6 y \quad \text { and } \quad \frac{\partial f}{\partial y}(x, y)=6 y^{2}-6 x .$$$\newline$(a) Determine all stationary points of $f$.$\newline$(b) Compute the second-order partial derivatives of $f$ and find the Hessian matrix $\boldsymbol{H}_{f}(x, y)$ of $f$. Determine the points in the $(x, y)$ plane where $f$ has a local maximum, a local minimum or a saddle point.$\newline$(c) It is now stated that $f(0,0)=1$. Determine the second-degree Taylor polynomial $P_{2}(x, y)$ for $f$ with the expansion point $f$$0$.$\newline$Exercise $2$. Define the function $f$$1$ by$\newline$$$f(x)=\left\{\begin{array}{ll} \frac{\sin (x)}{x} & x \neq 0, \\ 1 & x=0 . \end{array}\right.$$$\newline$(a) Find the third-degree Taylor polynomial $f$$2$ of $f$$3$ with expansion point $f$$4$.$\newline$(b) Show that$\newline$$$\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1 .$$$\newline$Hint: Use (a) and Taylor's limit formula.$\newline$(c) Argue that $f$ is continuous on $f$$6$.$\newline$(d) Compute, e.g. using SymPy, a decimal approximation of $f$$7$. You should include at least $5$ decimals.$\newline$(e) Compute a Riemann sum $f$$8$ approximating $f$$7$, where we require that $f$$0$ for each $f$$1$.