Q1: What is the correct formula for the Hessian of a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$?

A1: $$ H_f(x) = \begin{pmatrix} \frac{ \partial^2 f }{\partial x_1^2 }(x) & \frac{ \partial^2 f }{\partial x_1 \partial x_2 } (x)\ \frac{ \partial^2 f }{\partial x_2 \partial x_1 }(x) & \frac{ \partial^2 f }{\partial x_2^2 } (x) \end{pmatrix} $$

A2: $$ H_f(x) = \begin{pmatrix} \frac{ \partial^2 f }{\partial x_1^2 }(x) & \frac{ \partial^2 f }{\partial x_2 \partial x_2 }(x) \ \frac{ \partial^2 f }{\partial x_2 \partial x_2 }(x) & \frac{ \partial^2 f }{\partial x_2^2 }(x) \end{pmatrix} $$

A3: $$ H_f(x) = \begin{pmatrix} \frac{ \partial f }{\partial x_1 }(x) & \frac{ \partial f }{\partial x_2 }(x) \ \frac{ \partial f }{\partial x_2 }(x) & \frac{ \partial f }{\partial x_1 } (x) \end{pmatrix} $$

A4: $$ H_f(x) = \begin{pmatrix} \frac{ \partial^2 f }{\partial x_2^2 }(x) & \frac{ \partial^2 f }{\partial x_1 \partial x_2 } (x) \ \frac{ \partial^2 f }{\partial x_2 \partial x_1 } (x) & \frac{ \partial^2 f }{\partial x_1^2 } (x) \end{pmatrix} $$

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x_1, x_2) = x_1 + x_2^2 + x_1^4$. What is the second Taylor polynomial $T_2$ of $f$ with expansion point $(0,0)$?

A1: $T_2(h_1, h_2) = h_1 + h_2^2$

A2: $T_2(h_1, h_2) = h_1 + h_2$

A3: $T_2(h_1, h_2) = h_1^2 + h_2$

A4: $T_2(h_1, h_2) = h_1^2 $

A5: $T_2(h_1, h_2) = h_1^3 + h_2^2$

Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x_1, x_2) = \cos(x_1 x_2)$. What is the first Taylor polynomial $T_1$ of $f$ with expansion point $(0,0)$?

A1: $T_1(h_1, h_2) = 1$

A2: $T_1(h_1, h_2) = 1+ h_1 + h_2 $

A3: $T_1(h_1, h_2) = h_1 + \cos(h_2) $

A4: $T_1(h_1, h_2) = 1+ h_1 $

A5: $T_1(h_1, h_2) = h_2 $