# Calculus, constrained minimization and Lagrange multipliers: Is the optimal critical point a local minimizer?

## TL;DR

This paper examines the conditions under which critical points in constrained optimization are local minimizers, especially in low-dimensional cases, and discusses common misconceptions and counterexamples in undergraduate calculus.

## Contribution

It clarifies the sufficient conditions for local minimality and presents counterexamples to prevalent undergraduate misconceptions about Lagrange multipliers.

## Key findings

- Counterexamples to undergraduate statements about critical points and minimizers
- Conditions under which critical points are local minimizers in low dimensions
- Discussion of misconceptions in undergraduate calculus related to constrained optimization

## Abstract

In this short note, we discuss how the optimality conditions for the problem of minimizing a multivariate function subject to equality constraints have been dealt with in undergraduate Calculus. We are particularly interested in the 2 or 3-dimensional cases, which are the most common cases in Calculus courses. Besides giving sufficient conditions to a critical point to be a local minimizer, we also present and discuss counterexamples to some statements encountered in the undergraduate literature on Lagrange Multipliers, such as `among the critical points, the ones which have the smallest image (under the function) are minimizers' or `a single critical point (which is a local minimizer) is a global minimizer'.