Min Value of 1/6 (a^3 + b^3 + c^3 - 3abc) for Distinct Ints

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Homework Help Overview

The problem involves finding the minimum value of a specific expression involving distinct positive integers \(a\), \(b\), and \(c\), under the constraint that \(ab + bc + ca \geq 107\). The subject area pertains to inequalities and optimization in algebra.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to use the AM-GM inequality but expresses confusion about which numbers to apply. Some participants suggest the use of Lagrange multipliers as a potential method. Others mention achieving a solution using AM-GM, prompting requests for clarification on the approach.

Discussion Status

The discussion includes various approaches, with some participants exploring the use of inequalities and others sharing their experiences with reaching a solution. There is an ongoing exchange of ideas, and while some have found answers, others are still seeking guidance.

Contextual Notes

Participants are navigating the constraints imposed by the problem, particularly the requirement for distinct positive integers and the specific inequality condition.

NEILS BOHR
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Homework Statement


If a , b ,c r distinct +ve integers such that ab + bc + ca is greater than equal to 107 , then find the minimum value of 1/6 ( a^3 + b^3 + c^3 - 3abc )


Homework Equations





The Attempt at a Solution


tried usin AM GM equality but i m confused which nos. shud i use??
 
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This smells very much like Lagrange multipliers
 
got the ans finally using AM GM only...:biggrin:
 
NEILS BOHR said:
got the ans finally using AM GM only...:biggrin:
Can you show me?... i got stuck too. XD
 

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