(1) 9. Suppose f is continuous on [0, 1] with f(0) = f(1) which of the following statement(s) must be true?
(i) f is uniformly continuous on [0,1].
(ii) If f f 0 then f(x) = 0 for all x = [0, 1].
(iii) there exists c € (0, 1) such that f'(c) = 0.
9.
(1) 10. Let a,b R, a (i) If
C
is a number in between f'(a) and f'(b) then there exists c € (a,b) such that Y = f'(c).
(ii) There exists c E (a, b) such that f'(c)(b-a) = f(b) - f(a).
(iii) f is bounded on R if f' is bounded on R.
(1) 11. Which of the following function(s) is (are) integrable on [0,1].
=
(i) f(x)=
q
(ii) f(x)=
x #Q
=q>0 and ged(p,q) = 1.
if x= for some n ≥1
otherwise.
(iii) Same as (ii) except f(1/2) = 1/2.
10.
11.
(1) 12. Suppose f is a decreasing function and g is an increasing function from [0,1] to [0,1]. Which of the following statement(s) must be true?
(i) If in integrable.
(ii) fg is integrable.
(iii) fog is integrable.
12.