Optimization of Functions of One Variable

Zrinka Lukač ()
Additional contact information
Zrinka Lukač: University of Zagreb, Faculty of Economics & Business

Chapter 12 in Economic Analysis Through Mathematics, 2026, pp 257-295 from Springer

Abstract: Abstract 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.

Keywords: Local extrema; Global extrema; Intervals over which a function is increasing or decreasing; Stationary points; Inflection points; Convex and concave functions; The first derivative test; The second derivative test; Algorithm for determining local extrema (search for similar items in EconPapers)
Date: 2026
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sptchp:978-3-032-08812-3_12

Ordering information: This item can be ordered from
http://www.springer.com/9783032088123

DOI: 10.1007/978-3-032-08812-3_12

Access Statistics for this chapter

More chapters in Springer Texts in Business and Economics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().