Suppose that f : [0, 3] → R is continuous on [0, 3], differentiable on (0, 3) and f(0) = 0, f(1) = 1, f(3) = 1.

- Show that there exists c1 ∈ (0, 3) such that f'(c1) = 1.
- Show that there exists c2 ∈ (0, 3) such that f'(c2) = 0.
- Show that there exists c3 ∈ (0, 3) such that f'(c3) = 1/3

I don't even know where to start, I looked at Rolle's Theorem, Mean Value Theorem and another different theorems but I can't see anything. Even just one part of the solution would be great.

Edit: Typo. My teacher had a typo [intervals should be (0,3) not (0,2)] so I now know the third question can be showed by Mean Value Theorem.