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.