The polynomial function is defined for all real numbers by .
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
The polynomial function p is defined for all real numbers by p(x)=x4−4x3+10.
(a) Find all critical numbers of p.
(b) Evaluate the second derivative at each critical number and classify every critical point when possible.
(c) For any critical point at which the second derivative test is inconclusive, apply a suitable alternative method to determine whether the point corresponds to a local extremum. Justify your reasoning thoroughly.
(d) Determine the absolute maximum and absolute minimum values of p on the closed interval [0,5], supporting your answer with appropriate analysis.