Find an extremal of the functional $ \int_{0}^{1}(u'')^2dx - (u(0) - u(1))^2$.

Question

I need to find an extremal of the functional $I[u] = \int_{0}^{1}(u'')^2dx - (u(0) - u(1))^2 \\ $, without any boundary conditions. Here's my solution: $ \\ \frac{d I[u+th]}{dt}|_{t=0}=2u''h'|^{1}_{0} -2u'''h|^{1}_{0}+2(u(1)-u(0))h|^{1}_{0}+2\int_{0}^{1}u^{(4)}hdx.$ When$\ \ h\in \mathrm{C}^{\infty }[0, 1], h(0)=h(1). \\ $ By the equation, we get $2\int_{0}^{1}u^{(4)}hdx=0 \Rightarrow u(x)=\mathrm{C}_{1}\frac{x^3}{6}+\mathrm{C}_{2}\frac{x^2}{2}+\mathrm{C}_{3}x+\mathrm{C}_{4}. \\ $
And if $ h(0)=h(1) \Rightarrow u(1)-u^{(3)}(1)-u(0)+u^{(3)}=0\Rightarrow \mathrm{C}_{1} = 0, \mathrm{C}_{2}=2\mathrm{C}_{3}.\\ $
So the answer is $ u(x)=x^2+\mathrm{C}_{3}x+\mathrm{C}_{4} \\ $. Is it Correect?
Here is one more question, how to find extremal of functional $ \int_{0}^{1}udx$, using previous functional with Lagrange multiplier? Thanks in advance!