# Calculus Pop Quiz

Calculus Level 3

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.

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