Mathematics

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.

Intuitive explanation

The key insight is that a local minimum of $f'$ tells you about the concavity of $f$ , not directly about $f$ itself. Since $f'$ has a local minimum at $x=3$ , its derivative $f''(3)=0$ and $f''$ changes from negative to positive there—meaning $f$ transitions from concave down to concave up. This is an inflection point of $f$ , where the curvature flips, even though $f$ itself may not have a local extremum.

Intuitive explanation

Browse the problem library