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Unformatted text preview: x  kt x .
If t x this inequality is obvious. Suppose that t x. Using the mean value theorem we choose a
number c between t and x such that
ft fx
fc
tx
and observe that
ft fx
f c  k
tx
7. Given that f is a function defined on an interval S and that the inequality
f t f x  t x  2
holds for all numbers t and x in S, prove that f must be constant.
Given any number x S we deduce from the inequality
ft fx
0
t x 
tx
that holds for all t x and the sandwich theorem that
ft fx
lim
0.
tx
tx
Since f x 0 for every x in the interval S, the function f must be constant.
S 8. Suppose that f and g are functions defined on R and that f x g x and g x f x for every real
number x.
a. Prove that f x f x for every number x.
Quite simply, for each x
f x g x fx. 242 b. Prove that the function f 2 g 2 is constant.
If we define h f 2 g 2 then h 2ff 2gg 2fg 2fg 0 and so h is constant. 9. Given that f is differentiable on the interval 0, Ý and that f x
f x1 f x
as x Ý. Solution: Suppose that 0 and, using the fact that f x as x Ý, prove that as x Ý, choose a number w such that the inequality
holds whenever x f x 
w. We shall now show that the inequality
f x1 f x
holds whenever x w. Suppose that x w. Applying the mean value theorem to f on the interval x, x 1 we now choose a number c between x and
x 1 such that
f x1 f x
f c
x1 x
and we observe that
f x1 f x
f x1 f x
f c  .
x1 x
10. Given that f is continuous on a, b and differentiable on a, b , and that f x approaches a limit w R as
x a, prove that f must be differentiable at the number a and that f a w.
We need to show that
ft fa
w
ta
as t a . Suppose 0 and choose 0 such that the condition f x w  whenever
x
a, b and x a . Given any number t
a, b satisfying the inequality a t a we choose
x between a and t such that
ft fa
f x
ta
and conclude that
ft fa
w.
ta a x t a 11. Prove that if f is differentiable on an interval a, b and f a 0 and f b 0 then there must be at least
one number c
a, b for which f c 0. Hint: Look at the number at which the function f takes its minimum value.
We know from Fermat’s theorem that f does not have its minimum at a or at b and so f must have
its minimum at some number c between a and b. From Fermat’s theorem again we deduce that
f c 0.
12. Prove that if f is differentiable on an interval S then the range of the function f must be an interval. Hint: Use the preceding exercise.
This argument should now be an exact parallel to the Bolzano intermediate value theorem for
continuous functions.
13. Suppose that f is differentiable on the interval 0, Ý , that f 0 0 and that f is increasing on 0, Ý . Prove 243 that if
gx fx
x for all x 0 then the function g is increasing on 0, Ý . Solution: In order to prove that the differentiable function g is increasing on 0 for every number x 0. We therefore need to show that
xf x f x
0
x2
for all x 0 and we can express this desired inequal...
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 Fall '08
 STAFF
 Math, Calculus

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