ABSTRACT
We now turn to the problem of finding the maximum and minimum of a function on the boundary of an open set. The example of the set x2+y2 ≤ 1 that we gave earlier is in some sense typical. The open set in this case is x2 + y2 < 1 and its boundary is x2 + y2 = 1. In this particular case the boundary is the level set of a function and this occurs quite regularly in practice-in fact there is practically no real loss of generality in assuming that this is always the case. So we forget about the open set and just look at the problem of finding the maximum and the minimum of the sufficiently regular function of two variables f(x, y) on a level set of the sufficiently regular function g(x, y). We suppose, in our initial analysis, that the level set has the form
{ (x, y) ∈ U ; g(x, y) = 0} for some open set U in R2.
