Prove that the solution to the mixed boundary value problem

is the unique C2 function that minimizes the modified energy functional

when subject to the inhomogeneous boundary conditions. Hint: Mimic the derivation of Theorem 11.10.
Remark: Physically, the inhomogeneous Neumann boundary condition u'(„“) = β represents an applied strain at the free end, and contributes an additional term to the total energy of the mechanical system.

0. (11.106)