There is plenty of information on maximizing functions given some constraints. But what if I already have the function value and want to know the maximum possible value of an argument? If I know the surface, what is the highest point? I can't find the right keywords on this topic. Here is the general question:
What is the minimal/maximal value of $x$ that satisfies constraints $f(x,y,\ldots )=0, g(x,y,\ldots )=0, \ldots $?
P. S. It would be interesting to know general things, keywords and resources. But if it helps, I an currrently particulary interested in the case that the individual surfaces determined by these constraints are smooth and convex or flat.
Never mind. I spent two hours thinking and reading on this but it took only 5 minutes after posting to realize the answer. I won't delete the question in case anyone else gets stuck like me.
The solution: use the Lagrange multiplier as usual. The function to maximize is simply $x$.