Let be given functions with derivatives
satisfying
for all in some interval
which contains 0. (Note:
and
satisfy all of these conditions.)
- Prove that these functions must satisfy the Pythagorean identity
- If
and
are any other pair of functions satisfying all of these conditions prove that
- What else can we say regarding the functions
and
?
- Proof. First, we show that
for some constant, and then we will prove that constant must be 1. To show
is constant we take the derivative
This holds for all
; hence, by the zero derivative theorem (Theorem 4.7 (c) in Apostol) we must have
for some constant
for all
. Now, to show this constant
, we know by hypothesis that
and we have
Hence,
everywhere on
- Proof. Let
and
be another pair of functions satisfying the given relations:
Then define
This implies
Using the relations
, and
we have
for all
. Hence,
for some constant
. Then, we evaluate
and use the given relations
and
,
Hence,
for all
. Since it is a sum of squares (which must be nonnegative) we have that
if and only if
for all
. Hence we must have
- I’m not entirely sure what Apostol wants us to say here with respect to this solution. Since we have established that
and
satisfy these properties and that any functions satisfying these properties are unique, we can conclude that
and
are the unique functions which satisfy the given properties.