Let f be the polynomial function defined by f(x)=x4−4x3+10 on the closed interval [−1,4].
(a) Calculate f′(x) and determine all critical points of f in the interval [−1,4]. Show all algebraic work.
(b) Construct a complete sign chart for f′(x) on [−1,4] and use it to determine the intervals on which f is increasing and the intervals on which it is decreasing.
(c) Classify each critical point as a local maximum or a local minimum. Justify your answer.
(d) Evaluate f at every critical point found in part (a) and at the endpoints x=−1 and x=4. Identify the absolute maximum and absolute minimum values of f on [−1,4], and state where they occur.