Constrained problem

The solution of problem 4.3 is slightly more complicated. The presence of constraints reduces the feasible set of independent variables that are solutions of the problem. So the solutions, (i.e. the value of independent variables that minimize the objective function), must be searched in the set $x\in C\subset X$ that satisfies all the constraints.
The most important method to solve the problem of the minimization taking into account the satisfaction of some constraints (and, incidentally, the method most useful for our real problem) is the method of the Lagrange multiplier (and its derived, the method of the penalty function).