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
4 years, 5 months 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).

Staff - 4 years, 5 months ago

- 4 years, 1 month ago

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.

- 4 years, 5 months ago

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.

- 4 years, 5 months ago