According to world-famous efficiency expert Dr. I. E. Pertchart, the productivity P of a group of N engineers working on the same project is given by his productivity formula

P=(1+x_1)*(1+x_2)*...*(1+x_k).

Here the project has been split into k <= N independent subprojects with x_i of the engineers assigned to the i-th subproject, x_1+x_2+...+x_k=N. Over all possible splits of the engineers, what are the maximum and minimum values of P that can be achieved? Proof?