How to correctly answer ‘Is the optimal critical point a local minimizer?’ in Calculus courses

Abstract

This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient conditions for a critical point to be a local minimizer, we also present and discuss counterexamples for some statements that can be found in the literature of undergraduate Calculus related to Lagrange Multipliers, such as ‘between 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’.

Keywords:

• Calculus
• constrained minimization
• Lagrange multipliers
• critical point
• local minimizer
• global minimizer

Acknowledgments

We are thankful to the remarks and suggestions of the reviewers, which helped us to improve the presentation of this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 In this paper, everything related to minima can be replaced by maxima, as the upcoming remark 2.1 makes clear!