8. Proof of theorem 1.
We need the fundamental lemma of calculus of variations:
Lemma: Let f be a function in C[a,b] and assume that
for every
Then
Proof of theorem 1: Let y be an extremal and let
Then
is an admissible and competing function for
. Consider the functional
Since y is an extremal we must have
If we move the derivative inside the integral we get
This means that
for all
We integrate the second term by parts and get
![](https://www.larserikpersson.se/wp-content/themes/evol-one-theme/assets/images/chap3en/eq808.gif)
<>
Since
<>
![](https://www.larserikpersson.se/wp-content/themes/evol-one-theme/assets/images/chap3en/eq810.gif)
this implies that
We thus have
for all
If we finally use the lemma above we can conclude that
The proof is complete.