Suppose That F 0 2 R Is Continuous On 0 2 Differentiable On 0 2 And F 0 0 F

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.

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

Solution

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Yes, #3 is IVT and by MVT since the secant line has slope of 1/3

#2 is Rolle’s Theorem

the function has the same f(1)=f(3)=1, so the derivative is zero somewhere in (1,3)

#1) MVT says the derivative is 1 on (0,1) because the slope of that secant line is 1