  At the horizon, a static extremal black hole solution in N=2 supergravity in
four dimensions is determined by a set of so-called attractor equations which,
in the absence of higher-curvature interactions, can be derived as
extremization conditions for the black hole potential or, equivalently, for the
entropy function. We contrast both methods by explicitly solving the attractor
equations for a one-modulus prepotential associated with the conifold. We find
that near the conifold point, the non-supersymmetric solution has a
substantially different behavior than the supersymmetric solution. We analyze
the stability of the solutions and the extrema of the resulting entropy as a
function of the modulus. For the non-BPS solution the region of attractivity
and the maximum of the entropy do not coincide with the conifold point.
