The paper contains four theorems concerning first order necessary conditions for a minimum in nonsmooth optimization problems in Banach spaces: a Lagrange multiplier rule for a mathematical programming problem in which an infinite dimensional equality constraint is included in the constraints, a general maximum principle for nonsmooth optimal control problems with state constraints, and a kind of multiplier rule for mathematical programming problems which applies when only finitely many equality constraints are present but when the Lipschitz continuity assumptions are removed. A summary of relevant background results from analysis is provided.