Optimization of Functions of One Variable

One of the most important applications of differential calculus in both mathematics and economics is finding points at which a function reaches a local minimum or maximum. In economics, for example, it is important to determine the basket of goods that maximizes utility, the level of production that maximizes profit, the input of labor (or capital) that minimizes the costs for a predetermined level of production, the level of production that minimizes average production costs, etc. Therefore, in this chapter we examine the problem of determining extrema of functions of one real variable. First, we introduce the concepts of local and global minima and maxima and then we use the first derivative to characterize and find intervals over which a function is increasing or decreasing, which will help us characterize and find extrema. Thus, we will obtain the necessary and sufficient conditions for the existence of extrema using the first derivative. Since each local maximum is related to the concept of a concave function and each local minimum is related to the concept of a convex function, we will characterize the convexity and concavity of a twice differentiable function using the second derivative. We then derive sufficient conditions for the existence of extrema using the second derivative. This will help us develop an algorithm for determining all local extrema of any twice differentiable function.