1549901369-Elements_of_Real_Analysis__Denlinger_

326 Chapter 6 • Differentiable Functions

Solution. Let x, y E R If x = y, the desired inequ ality is true, since both
sides are 0. Hence, assume x =/:-y. Without loss of generality, assume x < y.
The function f(x) = sinx is continuous on [x,y] and differentiable on (x,y).
Hence, by the mean value theorem, 3 c E ( x, y) 3

f

, ( ) _ sin x - sin y..

c - , i.e.,

x-y

sinx - siny

COSC= ,SO

x - y

I I

I

cosc = sin x - sin y I.

x-y

B ut I cos c I ::; 1. H ence, I sin I x - sin I y I ::; (^1). Th ere £ ore,
x-y
I sinx - sinyl ::; Ix - YI· D
EXERCISE SET 6.4

1. Prove Theorem 6.4.1, Subcase 2b.


2. In each of the following, give a n example of a function that fits the given
   conditions and for which the conclusion of Rolle's theorem does not hold:
   (a) f is continuous on [a, b] and f(a) = j(b).
   (b) f is differentiable on (a, b) and f(a) = f(b).
   (c) f is continuous on [a , b] and differentiable on (a, b).


3. Prove that if f is differentiable on a nonempty interval I , and f'(x) is
   never 0 for x E I , then f must be 1-1 on I.


4. P rove that the converse of Exercise 3 is not valid, by showing a counterex-
   ample: a function f that is 1-1 on a nonempty interval I but f'(x) = 0
   for some x EI.


5. Prove that the function j(x) = 3x^5 - 2x^3 +12x - 8 is 1-1 on (- oo, + oo).


6. Prove that the function f(x) = x^3 + x^2 - 5x + 3 is 1-1 on [l , 5], but not
   on (-oo,+oo).


7. Prove that the equation 7 x^3 - 5x^2 + 4x - 10 = 0 has exactly one real root.
   [You must prove two things: that the equation has at least one real root,
   and that it cannot have more than one.]