I'm able to get this problem up to certain point, but then I doubt when adding the c's.

So by the Mean Value Theorem you have:

f'(c) = (f(b) - f(a))/(b-a)

the interval [-1,8] is your [a,b] so pluging in with your givens you should get f'(c) = 1

But after that I doubt my answer, I would suggest that c1=c2=c3, thus for k you get 3

But I will like some confirmation though.