Let g(x)=x4−4x3+6x2+1 be defined on the closed interval [−1,4].
(a) Determine all critical numbers of g in the open interval (−1,4).
(b) Evaluate g′′ at each critical number found in part (a) and classify each critical point as a local minimum or local maximum. Provide justification.
(c) Using the information from parts (a) and (b) together with the values of g at the endpoints, determine the absolute minimum and absolute maximum values of g on the interval [−1,4]. State each extremum and the x–value at which it occurs, and justify your answer.