This is a question that originated with World of Warcraft. I have a solution, but I don't know where to look up other problems of the same kind for a better explanation.


There is a plane with p axis and q axis.


There is a function f(x,y) = 2px+qy^2 such that f(x,y) is only defined on the curve x^3+y^3=1.


Claim: There are regions of the plane for which f has exactly one critical point and exactly three critical points.


The proposed solution was:
F(x,y,k)=2px+qy^2+k(x^3+y^3-1)


This was not explained, but the last term multiplies k by zero.
I think this is some kind of Lagrange multiplier method, but it's not like any use of that method that I've seen before. 


The solution continued:
Fx=2p+3kx^2=0
Fy=2qy++3ky^2=0
I assume that Fx means "derivative of F with respect to x" rather than a simple product.


This part of the solution looked like a Karusch-Kuhn-Tucker method, but I have only seen simple examples of that.


The most applicable book I have found for this kind of problem is Optimization Concepts and Applications in Engineering by Belegundu and Chandrupatla.


But perhaps this is not a KKT method specifically - perhaps it's just advanced calculus.


I have one book that touches on this kind of problem but doesn't really explain much. I want to find a book (or better yet, a free online wiki) that explains this kind of problem thoroughly.


Thanks



Is this a Karusch-Kuhn-Tucker method or something else?
It is using KKT. For details, I'd recommend the Convex Optimization book by Boyd and Vandenberghe. 


As a side note, this is not research level mathematics, and I would like to vote to close, but since I am still a new user, I don't know how to do it and I just leave this comment as an answer (can't comment yet either). 


