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Local and global extrema for functions of several variables

Published online by Cambridge University Press:  09 April 2009

Bruce Calvert  and
M. K. Vamanamurthy
Bruce Calvert
Affiliation:
University of AucklandAuckland, New Zealand
M. K. Vamanamurthy
Affiliation:
University of AucklandAuckland, New Zealand
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Abstract

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Let p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.

1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 3 , May 1980 , pp. 362 - 368
Copyright
Copyright © Australian Mathematical Society 1980

References

Cronin, J. (1964), Fixed points and topological degree in nonlinear analysis, Math. Surveys, No. 11 (Amer. Math. Soc., Providence, R.I.).Google Scholar
Milnor, J. W. (1965), Topology from the differentiable viewpoint (The University Press of Virginia, Charlottesville).Google Scholar