A family of twice–differentiable functions

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

Problem

\{k_m\}

, indexed by the real parameter

m

, has the following property for every fixed

m

: • The graphs of

k_m

and

k_m''

intersect exactly at the three points

(-1,0)

(0,m)

, and

(2,3)

. • The graph of

k_m'

has relative extrema only at

x=-1

and

x=2

, and

k_m'(0)=0

. •

k_m'

is positive on

(-1,2)

and negative elsewhere. •

k_m''

is continuous, changes sign only at

x=-1

x=0

, and

x=2

, and satisfies

k_m''(x)>0

for

x\in(-1,0)\cup(2,\infty)

. (a) For a fixed value of

m

, determine all possible sign patterns of

k_m''

that are consistent with the information above. Represent your answer in sign–chart form over the real line. (b) Prove that, for every

m

, the point

(0,m)

must be an inflection point of the graph of

k_m

. (c) Show that

m

must satisfy

m<3

. Justify rigorously using properties of

k_m

and its derivatives.