# A Surface with One Local Minimum

by J. Marshall Ash (DePaul University) and Harlan Sexton

Mathematics Magazine
May, 2008

Subject classification(s): Calculus | Several Variable Calculus | Number Concepts
Applicable Course(s): 3.3 Mainstream Calculus III, IV | 4.0 Advanced Mathematics

A smooth surface in $\Re^2$ or $\Re^3$ has one critical point that is a local, but not global minimum. Must that surface have another critical point? While the answer in the 2-D version is "yes", in 3-D the answer is "no", as illustrated by examples in this paper. The authors further analyze for which surfaces the answer is "yes" in 3-D.

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Capsule Course Topic(s):
Multivariable Calculus | Optimization