Mathematics

Problem

Let

f

be a continuous function on an open interval

I

that contains a fixed real number

a

. Let

h

be a differentiable function on

I

with

h(x)\neq a

for all

x\in I

. Define the accumulation function

G(x)=\int_{h(x)}^{a} f(t)\,dt, \qquad x\in I.

Answer the following questions about

G

. (a) Determine

G'(x)

in terms of

f

h

, and

h'(x)

. (b) Provide a clear, logical explanation for the appearance of the overall sign in your expression for

G'(x)

, making explicit reference to the change of limits in the definite integral. (c) State the hypotheses on

f

and

h

that guarantee the differentiability of

G

I

. Justify each condition you list.