Existence and Uniqueness of a Curve with Both Minimal Length and Minimal Area

Consider the family of generalized parabolas {y=−axr+c|a,r,c,x>0,risafixedconstant} that pass through a given point in the first quadrant (and hence, depend on one parameter only). Find the parameter values for which the piece of the corresponding parabola in the first quadrant either encloses a...

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Bibliographic Details
Published in:Mathematics (Basel) 2022-11, Vol.10 (21), p.4061
Main Authors: Fuxman, Ariel, Gul, Shai
Format: Article
Language:English
Subjects:
Algorithms
Analysis
Asymptotic properties
Convexity
Curves
Fixed points (mathematics)
Functions, Implicit
geometric optimization
implicit functions
Maxima and minima
Parabola
Parabolas
Parameters
Quadrants
Variables
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Summary:Consider the family of generalized parabolas {y=−axr+c|a,r,c,x>0,risafixedconstant} that pass through a given point in the first quadrant (and hence, depend on one parameter only). Find the parameter values for which the piece of the corresponding parabola in the first quadrant either encloses a minimum area, or has a minimum length. We find a sufficient condition under which given the fixed point, the area minimizing curve and the length minimizing curve coincide. The problem led us to a certain implicit function and we explored its asymptotic behavior and convexity.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10214061