The main result concerns a smooth map T of a Banach space X into itself which has an unstable fixed point \(x_ 0\). We prove that if the spectral radius \(\lambda_ 0\) of the Fréchet derivative of T at \(x_ 0\) is an eigenvalue which exceeds one and appropriate additional assumptions hold, then there is a smooth invariant curve emanating from \(x_ 0\) which might be called the ''most unstable manifold'' of \(x_ 0\). The curve is parametrized by a smooth function satisfying a functional equation involving T and \(\lambda_ 0\). This result is shown to be especially useful when the map T possesses certain monotonicity conditions. In this case, the curve can be shown to be monotone and to terminate on a stable fixed point of T.