How can I find the minimum / maximum of a function for one variable defined implicitly (f(x, y, z) = c) with a constraint g(x, y) = c on the domain?
For example, say you wanted to minimize for z:
$$-4\,x\,y^2\,z^3+2\,y^3\,z^2+x^3\,y^5+5\,x\,y^4=3$$
given that the xy coordinates lie on the unit circle:
$$x^2+y^2=1$$
How would this be done? Is it even possible?
Consider the function $F(x,y,z)=z$ and minimize it subject to the two constraints you have. Writing the constraints as $f=3$ and $g=1$, you want two Lagrange multipliers, $\lambda$ and $\mu$. Solve for where the constraints hold along with $$\nabla F =\lambda\nabla f + \mu\nabla g.$$