Functions and are differentiable through fourth order at

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Functions

f

and

g

are differentiable through fourth order at

x=a

. Their derivatives at

x=a

satisfy

f(a)=g(a)=0,\quad f'(a)=g'(a)=0,\quad f''(a)=g''(a)=0,\quad f'''(a)=6,\; g'''(a)=2.

Assume

g^{(4)}(a)

exists and is finite. (a) Explain why the information given about the first and second derivatives is insufficient to evaluate

\displaystyle \lim_{x\to a}\dfrac{f(x)}{g(x)}

. (b) Determine the smallest positive integer

n

such that the

n^{\text{th}}

derivatives of

f

and

g

x=a

guarantee a non–zero ratio. (c) Find

\displaystyle \lim_{x\to a}\dfrac{f(x)}{g(x)}

and justify your reasoning using the derivative data provided.