Mathematics

Problem

The derivative of a differentiable function

f

satisfies the following conditions: •

f'(x)=0

only at

x=-3

and

x=5

. •

f'(x)>0

for

-3<x<5

. •

f'(x)<0

for

x<-3

and for

x>5

. Which of the following accurately describes the behavior of

f

? a)

f

is increasing on

(-\infty,-3)

and

(5,\infty)

and decreasing on

(-3,5)

. b)

f

is decreasing on

(-\infty,-3)

and increasing on

(-3,5)

and

(5,\infty)

. c)

f

is increasing on

(-3,5)

and decreasing on

(-\infty,-3)

and

(5,\infty)

. d)

f

is decreasing on

(-\infty,-3)

, constant on

(-3,5)

, and increasing on

(5,\infty)

Intuitive explanation

The key is recognizing that the derivative tells you exactly where $f$ increases, decreases, and has turning points. Since $f'(x) > 0$ between $x = -3$ and $x = 5$ , the function climbs steeply there; where $f'(x) < 0$ on either side, it falls. The zeros of $f'$ at $x = -3$ and $x = 5$ mark a local minimum and maximum, respectively—the peaks and valleys of the original function.

Intuitive explanation

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