Mathematics

Problem

Let a differentiable function

f

have derivative

f'

satisfying •

f'(x)>0

for

x<-1

and

x>3

•

f'(x)<0

for

-1<x<3

At which set of

x

–values is

f

most likely to attain a local maximum and a local minimum, respectively? a) local maximum at both

x=-1

and

x=3

b) local maximum at

x=-1

, local minimum at

x=3

c) local maximum at

x=3

, local minimum at

x=-1

d) local minimum at both

x=-1

and

x=3

Intuitive explanation

The key is recognizing that local extrema occur where the derivative changes sign—not just where it equals zero. Since $f'$ tells you whether $f$ is increasing or decreasing, a local maximum happens when $f'$ switches from positive to negative, and a local minimum when it switches from negative to positive. Here, $f'$ changes sign at $x = -1$ and $x = 3$ , so identify which transition corresponds to which extremum type.

Intuitive explanation

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