Q1: Let $f: \mathbb{R} \rightarrow \mathbb{R}^2$ be a $C^1$-function. Which claim is correct?

A1: All the points $x \in \mathbb{R}$ are critical points of $f$.

A2: All the points $x \in \mathbb{R}^2$ are critical points of $f$.

A3: There are no critical points of $f$.

A4: There is only one critical point of $f$.

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the function given by $f(x) = x$. Which claim is correct?

A1: There are no critical points of $f$.

A2: All the points $x \in \mathbb{R}^2$ are critical points of $f$.

A3: There are no regular values of $f$.

A4: There is only one critical point of $f$.

Q3: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a $C^\infty$-function. What is correct formulation of the regular value theorem?

A1: If $c$ is a regular value of $f$, then $f^{-1}[{ c } ]$ is an $(n-m)$-dimensional submanifold of $\mathbb{R}^n$.

A2: If $f^{-1}[{ c } ]$ is an $(n-m)$-dimensional submanifold of $\mathbb{R}^n$, then $c$ is a regular value of $f$.

A3: If $f^{-1}[{ c } ]$ is a critical point of $f$, then $c$ is a regular value of $f$.

Q4: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x,y) = x^2 + y^2 - 1$. Is $f^{-1}[{ 0 } ]$ a submanifold?

A1: Yes, it is by the regular value theorem.

A2: No, 0 is not a regular value.

A3: No, the function is not well-defined.

A4: One needs more information.