(a) Find the extremal for the following:
i.e. satisfying
(1)
(b) For the extremal , show that
(2)
for all with .
(Hint: Multiplying equation (1) with , integrating in yields
(3)
and then integrating by parts in (3) with and .
(c) prove that is a minimizing curve.
(Hint: use inequality for any two real number . Then you use the identity (2).)