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Unformatted text preview: be Solution: Suppose that a and b are numbers in the interval S and a b. We shall show that
f a f b . Applying the mean value theorem to the function f on the interval a, b we choose a
number c between a and b such that
fb fa
fc
.
ba
Since f c 0 and b a 0 we deduce that f b f a 0 which gives us f a f b . c. Given that f is a function defined on an interval S and that f x 0 for every x S, prove that f must be
strictly decreasing on S.
Suppose that a and b are numbers in the interval S and a b. We shall show that f a f b .
Applying the mean value theorem to the function f on the interval a, b we choose a number c
between a and b such that
fb fa
.
fc
ba
Since f c 0 and b a 0 we deduce that f b f a 0 which gives us f a f b .
2. Suppose that f and g are functions defined on an interval S and that f x g x for every number x
Prove that there exists a real number c such that the equation
f x g x c
holds for every number x S. Hint: Apply the preceding exercise to the function f 241 g. S. 3. Suppose that f is continuous on an interval a, b and differentiable on the interval a, b and that f a f b .
Suppose that a c b and that f x 0 when a x c and f x 0 when c x b. Prove that f c is
the maximum value of the function f.
Since f is strictly increasing on the interval a, c we have f x f c whenever a x c and since f
is strictly decreasing on the interval c, b we have f c f x whenever c x b.
4. Given that f is a strictly increasing differentiable function on an interval S, is it true that f x must be positive
for every x S?
No. If we define f x x 3 for every number x then, although f is strictly increasing, we have
f 0 0.
5. Prove that if f is a differentiable function on an interval S and f x
must be oneone. 0 for every x S then the function f Hint: You should be able to make your conclusion very quickly from Rolle’s theorem.
0 for every number x in an interval S. Given numbers a and b in the interval,
We assume that f x
if a b and f a f b then we could apply Rolle’s theorem to find a number c between a and b
such that f c 0, which is impossible. Therefore f is oneone.
Note: Had this exercise told us that f x 0 for every x S we could have used Exercise 1 to deduce that f
is strictly increasing and if we had been told that f x 0 for each x then we would know that f is strictly
decreasing. Some students have wanted to argue that the given information, that f x
0 for every x S
guarantees that f x must either be always positive or always negative. This is true but we don’t know it yet.
The intermediate value theorem for derivatives is deduced in Exercises 10 and 11 below.
6. Given that f is differentiable on an interval S and that the function f is bounded on S, prove that f must be
lipschitzian on S.
Using the fact that f is bounded we choose a positive number k such that f x  k for every
number x S. Now suppose that t and x are any numbers in the interval S. We shall show that
f t f...
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 Fall '08
 STAFF
 Math, Calculus

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