MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let f(x)=x4−4x3+10 be defined on the closed interval [−1,4]. Answer the following.
(i) Find f′(x) and obtain all critical points that lie in [−1,4].
(ii) Compute f′′(x) and use it to classify each critical point, clearly identifying any point for which the result is inconclusive.
(iii) For every critical point at which the conclusion in part (ii) is inconclusive, determine the nature of the point by examining the sign change of f′(x) on suitably chosen intervals.
(iv) Using the information obtained, determine the absolute maximum and absolute minimum values of f(x) on [−1,4]. State the points at which they occur.