Let . Throughout, show all algebraic steps and clearly label each…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Let f(x)=3x6−5x4+2x3−7x+9. Throughout, show all algebraic steps and clearly label each derivative.
(a) Calculate f′(x), f′′(x), f′′′(x), and f(4)(x). Express each result as a simplified polynomial.
(b) Determine the least positive integer n for which f(n)(x) is identically zero for every real number x. Justify your answer by referencing structural features of f.
(c) Compute f(5)(2). Explain why your numerical result is consistent with your answer to part (b).
(d) Discuss the relationship between the degree of f and the sequence of non-zero derivatives found in parts (a)–(c).