A bar 1 meter long has stiffness c(x) = 1 +x at position 0 < x < 1. It is subject to an external force f(x) = 1 -x. The left end of the bar is fixed, while the right end is extended 1 cm.
(a) Write out and solve the boundary value problem governing the displacement of the bar.
(b) Write down the potential energy functional as the corresponding minimization principle.
(c) What is the energy of your solution?
(d) Construct two other possible displacement functions that satisfy the boundary conditions, and show, by direct computation, that they have larger potential energy.