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.