“…Moreover, after a relaxation result, a new necessary and sufficient condition for the existence of the minimum of F is introduced, which is expressed in terms of an upper bound for the assigned slope β−α b−a . On the other hand, a certain monotonicity property of the minimizers of free problems has been recently studied in [9] and [10], where it was proved that, under very mild assumptions, the competition set Υ can be restricted, without loss of generality, to those trajectories admitting at most one change of monotonicity. More precisely, if (P ) is solvable, then it admits a minimizer which is increasing in [a, x 0 ] and decreasing in [x 0 , b] (or vice versa) for some x 0 ∈ [a, b].…”