222 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6
(a) Use this theorem, together with the result of Exercise 8, Article 6.2, to prove
Rolle's theorem: Iff is continuous on [a, b], differentiable on (a, b), and if f(a) =
f (b) = 0, then there exists a point c E (a, b) such that f '(c) = 0.
(b) Consider the polynomial function f(x) = x4 + 3x + 1. Use Rolle's theorem to
prove that the equation f(x) = 0 has at most one real root between a = - 2 and
b = - 1. What theorem of elementary calculus guarantees that this equation has
at least one root between the given values of a and b?
(c) Use Rolle's theorem to prove the mean value theorem: If f is continuous
on [a, b] and differentiable on (a, b), then there exists a point c E (a, b) such
that f '(c) = (f (b) - f (a))/(b - a). [Hint: Apply Rolle's theorem to the function
F(x) = f(x) - G(x), where G(x) is the linear function determined by the points
(a, f(a)) and (b, f(b)). First, draw a picture and find the specific defining rule
for G(x).]
- (Continuation of Exercise 11) (a) Use the mean value theorem to prove that
iff' is identically zero on an interval I, then f is constant on I.
(b) Use the result in (a) to prove that iff' = g' on an interval I, then there exists
a constant c such that f = g + c on I.
(c) Use the mean value theorem to prove that iff' exists on an open interval
(a, b) and if f' is bounded on that interval [i.e., there exists M > 0 such that
I f'(x)l I M for all x E (a, b)], then f is uniformly continuous on (a, b) (recall
Exercise 7, Article 4.3).
- Frequently in mathematics, rather than proving that a certain set of properties
is satisfied uniquely by a given object, the best we can do is prove that the objects
satisfying the given conditions fall into certain very specific categories. Theorems
of this type are usually called classijication theorems. One example of a classification
theorem from elementary calculus involves the notion of antiderivative, or indefinite
integral. Recall that F is an antiderivative, or indefinite integral, of a function f
on an interval I, denoted F = j f(x)dx, if and only if Ft(x) = f(x) for all x E I.
(a) Prove that if F and G are both indefinite integrals of a function f on an inter-
val I, then F and G differ by some constant on I.
(b) Prove that if y = f (x) satisfies the differential equation y' + ay = 0 for all x E R,
then y = Ax) = ce-"" for some constant c. (Hint: Multiply the given equation
by e"".)
(c) Prove that there exists a unique function y = f(x) satisfying both the differ-
ential equation y' - 3y = 0 on R and the initial condition y(0) = 5.
//---
The subject of diflerential equations, from which (b) and (c) are simple examples, is
an area of mathematics in which the dual themes of existence and uniqueness are
particularly prominent. -
6.4 Preview of Additional Advanced
Methods of Proof (Optional)
In the first part of this text we have attempted systematically to lay a
foundation by which you might more easily and quickly be able to obtain
a measure of competence in reading and writing proofs, an important aspect
