(i) Write down the Euler-Lagrange equations for the volume integral

$$\int_{V}(\nabla u \cdot \nabla u+12 u) d V$$

where $V$ is the unit ball $x^{2}+y^{2}+z^{2} \leqslant 1$, and verify that the function $u(x, y, z)=x^{2}+y^{2}+z^{2}$ gives a stationary value of the integral subject to the condition $u=1$ on the boundary.

(ii) Write down the Euler-Lagrange equations for the integral

$$\int_{0}^{1}\left(\dot{x}^{2}+\dot{y}^{2}+4 x+4 y\right) d t$$

where the dot denotes differentiation with respect to $t$, and verify that the functions $x(t)=t^{2}, y(t)=t^{2}$ give a stationary value of the integral subject to the boundary conditions $x(0)=y(0)=0$ and $x(1)=y(1)=1$.