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 ;)