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log a.
For each x we have
f x exp x log a
and so
f x exp x log a
2. Given that a is a positive number and that a log a a x log a. 1, and given that f x log a x for every x 0, prove that
1
fx
x log a for every number x 0.
For each x 0 we have
af x x
which gives us
log a f x log x. Therefore
f x log a log x
and so
fx log x
log a from which the desired result follows at once.
3. Given that f and g are differentiable functions and f is positive, use the fact that
f x g x exp g x log f x
for each x to find a formula for the derivative of the function f g .
The value of the derivative of f g at a given number x is 256 exp g x log f x g x log f x gx
fx
fx fx gx g x log f x g x f x gx 1 f x. f
4. Given that f is differentiable on R, that f 0 1 and that f f, prove that the function exp is constant and
then conclude that f exp.
Since
f exp f exp
f exp f exp
f
0
exp
2
exp 2
exp
f
the function exp must be constant. To see that the constant is 1 we observe that
f0
1.
exp 0
5. Suppose that f : R R and that f x f x for every real number x. a. Given that g f f, prove that g g and deduce that there exists a real number a such that
g x 2ae x for every number x.
We see that
g f f f f g.
From Exercise 4 we know that the function g / exp is constant. If we call this constant 2a then
we have g x 2ae x for every number x.
b. Given that h x f x e x for every real number x, prove that the equation h x 2ae 2x holds for every
number x and deduce that there is there is a number b such that the equation
f x ae x be x
holds for every real number x.
For every number x we have
h x f x e x f x e x g x e x 2ae 2x .
It follows that the function whose value at every number x is h x ae 2x has a zero derivative at
every number and must therefore be constant. We call this constant b. Thus we have
h x ae 2x b
for every x, in other words
f x ae x be x .
6. Suppose that f : R R and that for all numbers t and x we have
f tx f t f x . a. Prove that either f x 0 for every real number x or f x
0 for every real number x.
Suppose that there exists a number at which the function f is zero. Choose such a number and
call it c. Given any number x we have
f x f x cc f x c f c 0
and so f is zero everywhere.
b. Prove that if f is not the constant zero, then f 0 1 and that f x 0 for every number x.
We suppose that f is not the constant zero. From the equation
f 0 f 00 f 0 f 0
and from the fact that f 0
0 we deduce that f 0 1. Finally, if x is any real number then the
equation
2
fx f x x f x
2
2
2
that f x 0. 257 c. Prove that if f is not the constant zero and if a f 1 then for every rational number x we have f x a x .
Deduce that if f is continuous on the set R then the equation f x a x holds for every real number x.
Compare this exercise with the last few exercises on continuity.
This exercise follows in exactly the same way as that earlier sequence of exercises.
7. Suppose th...
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 Fall '08
 STAFF
 Math, Calculus

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