Let f(x)=x3−6x2+9x+4 be defined on the closed interval [−1,5].
(a) Calculate f′(x) and determine all x–values in the interval [−1,5] at which f′(x)=0. These values are the critical points that lie in the interior of the interval.
(b) For each interior critical point found in part (a), determine on which side of the point f is increasing and on which side it is decreasing. Show all analytical work that supports your conclusions.
(c) Classify each interior critical point as a relative maximum, a relative minimum, or neither, and state the corresponding coordinate pair (x,f(x)) for each point.
(d) Evaluate f at the endpoints x=−1 and x=5. Indicate whether either endpoint represents a relative extremum on [−1,5], and justify your answer.