Let be a continuous function defined on the interval with the following properties

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

f

be a continuous function defined on the interval

[-3,5]

with the following properties. •

f(x)>0

for

-3<x<-1

f(x)=0

x=-1

, and

f(x)<0

for

-1<x<2

. •

f(x)=0

x=2

, and

f(x)>0

for

2<x<5

. •

f

is increasing on

(-3,-1)

, decreasing on

(-1,2)

, and increasing again on

(2,5)

. A particular solution

F

of the differential equation

\frac{dF}{dx}=f(x)

satisfies

F(-3)=4

. Answer the following questions about

F

. (a) Determine the intervals on

[-3,5]

where

F

is increasing and the intervals where

F

is decreasing. Give a brief explanation for each interval you identify. (b) Determine the intervals on

[-3,5]

where the graph of

F

is concave up and the intervals where it is concave down. Justify your answer. (c) Sketch a possible graph of

F

on the given interval, clearly indicating the point

(-3,4)

and all critical points and possible inflection points.