Let f(x)=x3−3x2+2 be defined on the interval [1,4]. Answer the following questions.
(i) Compute the average rate of change of f on [1,4].
(ii) Find all real numbers c in (1,4) for which f′(c) equals the average rate obtained in part (i).
(iii) Verify that every value of c obtained in part (ii) indeed lies within (1,4).