Introduction

Now we shall demonstrate how the inequalities that were derived in the preceding chapter can be used to treat an important and fascinating set of problems. These are maximization and minimization problems.

In the study of algebra and trigonometry, the problems you encountered were all of the following general nature: Given certain initial data, and certain operations that were to be performed, you were to determine the outcome. Or, conversely, given the operations that were performed and the outcome, you were to determine what the initial data must have been.

For example, you met three workmen, eager and industrious A, plodding and conscientious B, and downright lazy C. They were put to work digging ditches, constructing swimming pools, or building houses, and the problem was always that of determining how long it would take the three of them together to perform the job, given their individual efficiency ratings—or, given the time required for all three together, and the rates of effort of two of the three workers, it was required to find the rate of the third.

Sometimes you were given triangles in which two sides and an included angle were known, and sometimes you were given three sides of the triangle. In each case, the problem was that of determining the remaining sides and angles.