Let be the polynomial function defined by for all real numbers .
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let f be the polynomial function defined by f(x)=x4−4x3+10 for all real numbers x.
(a) Derive an explicit expression for f′(x). Use correct notation.
(b) Identify all critical numbers of f and state their exact numerical values.
(c) Determine each open interval on the real number line where f is increasing and each open interval where f is decreasing. Show analytical justification for every interval you list.
(d) For every critical number found in part (b), classify the corresponding point on the graph of f as a relative maximum, a relative minimum, or neither. Provide full reasoning based on your results from part (c).