A special class of nonlinear eigenvalue problems which exhibit branching to the left of. (below) the smallest eigenvalue is considered. This particular class serves to illustrate a procedure for bounding the leftward extent of a nonnegative solution branch. The procedure relies upon a well-known result about upper and lower solutions associated with a monotone operator. In the situation of left-branching bifurcation, the more difficult determination of a suitable lower solution is achieved by using an explicit solution to a simpler nonlinear problem. A physical example relative to the buckling of a nonlinearly elastic rod is worked out in detail.