Q1: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a function. What is the difference between ‘differentiable’ and ‘continuously differentiable’?

A1: There is no difference.

A2: A differentiable function is always a continuous differentiable function but not vice versa

A3: A continuous differentiable function is a differentiable function where $f^\prime$ is also continuous.

A4: A continuous differentiable function is a differentiable function where $f^\prime$ is also differentiable.

Q2: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a two-times differentiable function. Is $f$ continuously differentiable?

A1: Yes!

A2: No!

A3: One cannot say it.

Q3: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = x^5$. Which claim is not correct?

A1: $f$ is two-times differentiable.

A2: $f$ is 5-times differentiable.

A3: $f$ is 6-times differentiable.

A4: $f$ is $\infty$-times differentiable.

A5: $f$ has a local maximum at $x_0 = 0$.

Q4: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = (x-1)^2$. Which claim is correct?

A1: $f^{\prime \prime}(0) > 0$ implies there is a local minimum at $x_0 = 0$.

A2: $f^{\prime \prime}(1) > 0$ implies there is a local maximum at $x_0 = 1$.

A3: $f^\prime(1) = 0$ and $f^{\prime \prime}(1) > 0$ imply there is a local minimum at $x_0 = 1$.

A4: $f^\prime(1) = 0$ and $f^{\prime \prime}(1) < 0$ imply there is a local minimum at $x_0 = 1$.