Let f(x)=x4−4x3+10 for x in the closed interval [−1,3].
(a) Find all critical numbers of f that lie in (−1,3) and verify that no other critical numbers exist on the interval.
(b) Compute f′′(x) and evaluate f′′ at each critical number found in part (a). Classify each critical number as a local maximum, local minimum, or neither based on this information.
(c) Compute f(−1) and f(3).
(d) Compare the function values obtained in parts (a)–(c) to determine the absolute maximum and absolute minimum values of f on [−1,3]. State each extremum together with the x–value where it occurs and justify your conclusion.