A student is told that a polynomial of degree satisfies q^ (4) (x)=120 for all real x
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A student is told that a polynomial q of degree 4 satisfies
q(4)(x)=120for all real x,
with q(0)=−1. The student claims that these two pieces of information uniquely determine q.
(a) Explain why q(4)(x) must be a constant when q is a fourth–degree polynomial.
(b) Write the most general quartic polynomial consistent with q(4)(x)=120. Express your answer in terms of undetermined constants.
(c) Determine the number of additional, independent conditions needed to specify q uniquely. Justify your answer.
(d) Provide one possible set of additional conditions and use them to find the unique polynomial q(x) that meets all conditions you have specified.