Calculus Pop Quiz

Calculus Level 3

See how well you understand differentiability!

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

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

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

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

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

5) f(x)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 ;)

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