See how well you understand differentiability!
Consider a differentiable function f:R→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:R→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 ;)