Attainable region analysis has been used to solve a large number of previously unsolved optimization problems. This paper examines its relationship to Pontryagin's maximum principle and highlights the similarities and differences between the methods. It is shown that not all problems that can be solved by attainable region analysis are readily formulated as maximum principle problems. The fundamental process of mixing seems to be the main reason for this difference and the consequences of this are highlighted. The class of problems that can be formulated as maximum principle problems are then examined and the relationship between the two methods highlighted. Here, the maximum principle gives rise to a set of results that are very useful for finding an attainable region. In fact, from these results and the experience gained by solving AR problems, postulates on the nature of the boundary of the attainable region are proposed. Previous to this work the construction of the attainable region required a trial and error approach and the region thus generated was tested using the necessary conditions. These postulates should allow a more constructive approach to finding the attainable region boundary.