Calculus Pop Quiz

See how well you understand differentiability!

Consider a differentiable function $f : \mathbb{R} \rightarrow \mathbb{R}$ with $f(0)=0$ and $f'(0)=1$. How many of the following statements are true for any such function $f$?

1) $f(x)>0$ on $(0,q)$ for some positive $q$,

2) $f(x)$ is increasing on $(p,q)$ for some negative $p$ and some positive $q$

3) $|f(x)|$ is continuous,

4) There exists a differentiable function $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $g''(x)=f(x)$, and

5) $f'(x)$ is continuous.

I recently gave this pop quiz in week three of an introductory calculus class; it did not go so well. Let's see whether the "Brilliant class" can do better ;)