SLIGHT CORRECTION/CLARIFICATION/AMPLIFICATION: Change: Now such a CC necessarily contains a critical point of f(x,y) inside it, because f(x,y)=0 on the closed circle-like curve CC and is (wlog) >0 everytwhere that is infinitesimally inside it, and is continuous and differentiable, hence must have a maximum on the compact region inside CC, which necessarily is >0, and which necessarily is a critical point. Q.E.D. To: Now such a CC necessarily contains a critical point of f(x,y) inside it, because f(x,y)=0 on the closed circle-like curve CC and (wlog) f>0 everywhere that is just infinitesimally inside CC -- i.e. within epsilon of CC and inside CC, for all epsilon<eps0, for some positive eps0; we here are taking advantage of the known fact |grad f|>0 everywhere on CC and continuity of grad f since f is polynomial, which forces (grad f).(inward normal vector to c)>delta>0 everywhere on CC for some delta>0 -- and f is necessarily bounded, continuous and differentiable throughout the interior of (and on) CC, hence must have a maximum on the compact region inside-and-on CC, which necessarily is >0 and strictly inside CC, and which necessarily is a critical point. [We are here using the well known theorem that any continuous bounded function on a compact set has a max; and the other well known theorem that if the function also is differentiable, then grad f =0 at any max and any min. Also we'd previously implicitly used the same theorem in 1 less dimension using the fact CC itself is a compact 1-dimensional set and grad f is continuous and differentiable. Q.E.D. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)