I just started learning calculus of variations, and got this problem on the first day:
find the and that minimizes .
My approach so far:
First find and , then integrate .
Define the answer as a new function and find the point where its partial derivatives are both zero:
This turns out to be a lot of work, and it's hard not to make any mistakes along the way. So I was wondering if there's a better way to get the solution. A shortcut, if you will.