For both the following statements, determine whether it is true or false. If it is

true, prove it. If it is false, give a counter example to disprove it.

(a) If f and g are positive and continuous functions on [1,) such that both

Z

1

f(x) dx and Z

1

g(x) dx diverge, then so does Z

1

f(x)g(x) dx.

(b) If f is a continuous function on (0,) such that the improper integral Z

1

f(x) dx

converges, then the improper integral Z

0

f(x) dx converges as well.