MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Let p be the polynomial function defined by
p(x)=4x4−x3+3x2−5.
(a) Derive an explicit expression for p′(x) and determine every real critical number of p.
(b) Construct a complete sign chart for p′(x). Using that chart, state the intervals on which p is increasing and those on which p is decreasing.
(c) Classify each critical number found in part (a) as a relative maximum, relative minimum, or neither. Support your conclusions with evidence from part (b).
(d) Compute p′′(x). Identify all points where p′′(x)=0 and determine which of these points are inflection points of the graph of p. Justify your answer.