A tool that is missing from the standard equipment of the actuarial student is the technique of maximizing, or minimizing, a function of several variables which are subject to certain constraints. The conception of an extreme value of a function, subject to constraints, is, we feel, as important as the mathematical technique of evaluating such a value in a particular instance.

We shall first deal with the technique and its mathematical development and then demonstrate some applications. It is as if we were demonstrating a new car, how it works and how to drive it. The first example is simple and is in the nature of a trial run to show how the car handles. Succeeding examples go further and we hope that the actuarial countryside into which we go will prove as interesting as the mechanical means of getting there. To many the vehicle itself will be familiar, but perhaps some of our byroads may lead to unfamiliar places or to familiar places by unfamiliar routes.