Let and be twice–differentiable functions on an open interval

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let

f

and

g

be twice–differentiable functions on an open interval

I

. Assume that •

f(x)>0

and

g(x)>0

for all

x\in I

, •

f

is increasing and concave up on

I

, and •

g

is increasing and concave down on

I

. Define the function

h

I

h(x)=\dfrac{f(x)}{g(x)}

. (a) Without computing any derivatives of

h

, describe two distinct ways in which the behaviors of

f

and

g

could influence the concavity of

h

. Your answer should reference the relative sizes or signs of derivatives of

f

and

g

rather than numerical values. (b) Suppose that at a point

x=a

I

the following conditions hold:

f'(a)>0

g'(a)>0

f''(a)>0

, and

g''(a)<0

. Based solely on these local derivative conditions and your reasoning in part (a), indicate whether

h

is expected to be concave up, concave down, or if the information is insufficient to decide. Justify your conclusion in complete sentences.

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