Consider the boundary value problem -u" = x, u(0) = u(1) =0.
(i) Find the solution.
(ii) Write down a minimization principle that characterizes the solution.
(iii) What is the value of the quadratic energy functional on the solution?
(iv) Write down at least two other functions that satisfy the boundary conditions and check that they produce larger values for the energy. (Use numerical integration to evaluate the integrals if necessary.)