Mathematics

Problem

A continuous function

f

is defined on the interval

[2,10]

. An associated accumulation function is defined by

F(x)=\int_{2}^{h(x)} f(t)\,dt,

where

h

is differentiable on

[0,3]

and satisfies

h(0)=2

and

h(3)=8

. (a) Express

F'(x)

in terms of

f

and

h

. Fully justify your result, including any required conditions on

f

and

h

. (b) Suppose

f(t)>0

for all

t

(2,10)

and

h

is increasing on

(0,3)

. Explain what can be concluded about the sign of

F'(x)

for

x

(0,3)

. Provide reasoning in the context of the definitions of

F

and

h

. (c) Using the information in part (b) together with the given endpoint values of

h

, determine whether

F

attains its absolute maximum or absolute minimum on

[0,3]

x=0

or at

x=3

. Justify your answer completely.