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Extremes of a function

Hello everyone. This is my first discussion, and I want to discuss the methods of finding the maximum and minimum value of a given function. I know of three important methods to do so, namely

AM GM HM Inequality

Maxima and Minima Concept from Calculus

Cauchy Schwarz Inequality (useful for trigonometric functions)

I would be glad to know if anyone has any other methods of finding the extremum values of a function. I am also interested to know/hear of problems related to such concepts.

Note by Rohan Rao
3 years, 1 month ago

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It would be helpful to separate between classical inequalities (like AM-GM, Cauchy Schwarz, Chebyshev, Rearrangement, etc) and calculus approaches (like first derivative, Lagrange Multiplier, Linear Programming, etc). Calvin Lin Staff · 3 years, 1 month ago

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I definitely want to learn more about these methods. Do tell me about them Krishna Ar · 2 years, 9 months ago

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If you're in doubt to find an equality case try putting some of the variables equal, 1, or 0. Most cases that will solve the problem. The only exception to that that I can remember is that recent USAMO problem about sqrt(x-1)+sqrt(y-1)+sqrt(z-1) which had a really strange equality condition. Faraz Masroor · 3 years, 1 month ago

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@Faraz Masroor I have another exceptional case, which has also been featured as a Brilliant problem. Maximize (a^2)b+(b^2)c+(c^2)a subject to the constraint a+b+c=1. Allowing all variables to be equal gives 1/9, which is incorrect. Rohan Rao · 3 years, 1 month ago

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