Q1: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function for which all partial derivatives of second order exist. How many second-order partial derivatives do we have?

A1: $4$

A2: $1$

A3: $2$

A4: $3$

Q2: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2$. What is the partial derivative $\frac{\partial^2 f}{ \partial x_1 \partial x_2 }(x)$

A1: $2$

A2: $1$

A3: $0$

A4: $3$

Q3: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2$. What is the partial derivative $\frac{\partial^2 f}{ \partial x_2^2}(x)$

A1: $0$

A2: $1$

A3: $2$

A4: $3$