Let be a twice–differentiable function on an open interval containing

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

Problem

Let

f

be a twice–differentiable function on an open interval containing

x=c

, and suppose

f'(c)=0

and

f''(c)>0

. (1) Explain geometrically how the sign of

f''

determines concavity of the graph of

f

near

x=c

. Include a discussion of the tangent line slopes on either side of

c

. (2) Using your explanation from part (a), provide a reasoned argument that

x=c

must correspond to a relative minimum of

f

. Your argument should rely only on the behavior of

f'

and the concavity information, not on any specific example. (3) Give an explicit example of a differentiable function

p

and a critical point

x=d

for which

p''(d)=0

yet

x=d

is not a relative extremum. Justify why this example illustrates that knowledge of

p''(d)

alone is sometimes insufficient for classification.