Let h(x)=x4−4x3+6x2+1 for −1≤x≤4. Show all work to support your conclusions.
(a) Compute h′(x) and determine the open intervals of increase and decrease of h on the given domain.
(b) Compute h′′(x) and determine the open intervals on which h is concave up and concave down.
(c) Identify all open intervals on which h is increasing while concave down, and explain how you arrived at your answer.
(d) Using your results from parts (a) and (b), determine the absolute minimum value of h on the closed interval [−1,4], and justify that your answer is indeed the absolute minimum.