That looks correct to me. Why do you think it is wrong? After all, the question says "If possible, find the stationary point", and your answer is that it is not possible, unless α=1.
Okay, I have worked this problem a few times and I end up with a solution that I don't think is correct. Any feedback would be awsome.
If possible, find the stationary point of the functional
So I use the Euler-Lagrange Equation
So now I end up with
I don't think this is correct and it would only be consistant with the boundry values if