Q1: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x_1, x_2) = 4$ for all $(x_1, x_2) \in \mathbb{R}^2$. Does $f$ has a local maximum at the point $(1,3)$?

A1: Yes!

A2: No, but it’s an isolated local minimum.

A3: No, the function has no local maxima.

A4: One needs more information.

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x_1, x_2) = x_1^2 + x_2^2 $ for all $(x_1, x_2) \in \mathbb{R}^2$. Does $f$ has a local maximum at the point $(0,0)$?

A1: Yes!

A2: No, but it’s an isolated local minimum.

A3: No, the function has no local maxima.

A4: One needs more information.

Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x_1, x_2) = \cos(x_1 x_2)$. What is the Hessian at $(0,0)$?

A1: $ \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}$

A2: $ \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix}$

A3: $ \begin{pmatrix} -1 & 0 \ 0 & 2 \end{pmatrix}$

A4: $ \begin{pmatrix} 1 & 2 \ 0 & 2 \end{pmatrix}$

A5: $ \begin{pmatrix} 1 & 0 \ 0 & 2 \end{pmatrix}$