Summary: 18.014ESG Problem Set 8
Pramod N. Achar
1. Suppose f : [a, b] R is continuous; furthermore, suppose that it is
differentiable on (a, b). Show that if Df(x) > 0 for all x (a, b), then
f is strictly increasing. (Hint: If f were not strictly increasing, use the
Mean-Value Theorem to find a point where Df is zero or negative.)
2. Exercises 19 and 21 from Section 5.5 of Apostol. Both of these exercises
involve computing derivatives of functions defined in terms of integrals.
But be careful--you cannot apply the First Fundamental Theorem directly
to either of them.
3. Prove that there exists at least one positive number a such that cos a = 0.
(Hint: Suppose that cos x = 0 for all x > 0. Show that cos x would have to
be positive for all x > 0. Then show that sin x would be strictly increasing
for positive x. It follows that if x > 0, then
0 = sin 0 < sin x < sin 2x = 2 sin x cos x.
From this, derive the inequality (2 cos x - 1) sin x > 0 for x > 0. Then,
show that cos x > 1/2 for x > 0. It follows that