Let f be the function defined by f(x)=x4−4x3+10 for −1≤x≤5.
(a) Compute f′(x) and determine all critical numbers of f on the closed interval [−1,5]. Show all work.
(b) Compute f′′(x) and determine the intervals on which the graph of f is concave up and concave down. Support each conclusion with appropriate justification.
(c) Find the coordinates of every point of inflection of the graph of f on [−1,5]. Justify that each point satisfies the required conditions.
(d) Using your results from parts (a)–(c), classify each critical number from part (a) as a location of a relative maximum, a relative minimum, or neither. Provide reasoning based on concavity.
(e) Determine the absolute maximum and absolute minimum values of f on [−1,5], and state where they occur.