A function is twice differentiable and satisfies for for

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A function

g

is twice differentiable and satisfies

g'(x)>0

for

x<2

g'(2)=0

, and

g'(x)>0

for

x>2

. In addition,

g''(2)=0

and

g''

changes sign at

x=2

. Which statement best describes the behavior of

g

x=2

? (a)

g

has a local maximum at

x=2

because

g'(2)=0

and

g''

changes sign. (b)

g

has neither a local maximum nor a local minimum but does have a point of inflection at

x=2

.},{ (c)

g

has a local minimum at

x=2

because

g''

changes sign from negative to positive. (d) No conclusion can be drawn since

g''(2)=0

provides insufficient information.