Question $7$

Exercise $3\mathrm{.}4\mathrm{.}10\mathrm{.}2$ Obtain the extremals $($if they exist$)$ of the following problem of Lagrange with functional, auxiliary conditions and boundary conditions as indicated. Ensure that the problem, as it is stated makes sense as a problem in the variational calculus:

$I[x]={\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}[(x_2^{}{)}^2-(x_2{)}^{2}dt,]$

$x_1^{}+x_2^{}+x_3^{}=0,x_1^{}+2x_3^{}=0,$

$x_1(0)=x_2(0)=x_3(0)=0,$

$x_1(\frac{}{2})=-,x_2(\frac{}{2})=3\frac{}{2},x_3(\frac{}{2})=\frac{}{2.}$