Let $F(u) = \int_0^1(u'')^2+u^2dx $ on $C^2[0,1]$ satisfying $u(0)=a,u(1)=b,u'(0)=c,u'(1)=d$ where $a,b,c,d \in \mathbb{R}$.
If $u_*$ is a minimizer, for $\phi \in C^2[0,1],\ \frac{d}{ds}| _{s=0} F(u_* +s\phi) = \int_0^1 (2(u'')\phi''+2u\phi) dx$
Im having trouble understanding what is the correct choice of "directions" $\phi$ in the derivations of the euler-lagrange eq's.
Any help would be appreciated,
Thanks