Suppose is twice differentiable and its first derivative has a local minimum at

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

Problem

Suppose

f

is twice differentiable and its first derivative

f'

has a local minimum at

x=3

. No other critical points of

f'

occur near

x=3

. Which of the following statements about the graph of

f

x=3

is most accurate? (a)

f

has a relative minimum at

x=3

because

f'(3)=0

and

f''(3)>0

. (b)

f

is concave up at

x=3

, but

x=3

is not necessarily an inflection point of

f

. (c)

f

has an inflection point at

x=3

where the concavity changes from concave down to concave up. (d)

f

has an inflection point at

x=3

where the concavity changes from concave up to concave down.