  In this paper, we study some problems with continuously differentiable
quasiconvex objective function. We prove that exactly one of the following two
alternatives holds: (I) the gradient of the objective function is different
from zero over the solution set and the normalized gradient is constant over
it; (II) the gradient of the objective function is equal to zero over the
solution set. As a consequence, we obtain characterizations of the solution set
of a quasiconvex continuously differentiable program, provided that one of the
solutions is known. We also derive Lagrange multiplier characterizations of the
solutions set of an inequality constrained problem with continuously
differentiable objective function and differentiable constraints, which are all
quasiconvex on some convex set, not necessarily open. We compare our results
with the previous ones. Several examples are provided.
