Question 1026930
Let {{{F =  x^2 + y^2 - 4x + 2y}}}.

Finding the critical points of F: 
{{{F[x] =  2x - 4 = 0}}}==> x = 2, and
{{{F[y] =  2y + 2 = 0}}}==> y = -1.
==> critical point is (2,-1).
Also, {{{F[xx] = 2 > 0}}}, {{{F[yy] = 2 > 0}}}, and {{{F[xy] = F[yx] = 0}}}
Implement the 2nd derivative test for two variables:
{{{F[xx]*F[yy] - (F[xy])^2 = 2*2 - 0^2 = 4 > 0}}}

==> There is local min at (2,-1).  Since it is the only critical point in the domain of the function (which is infinite open), it is also an absolute minimum.
The temperature of the coldest point is thus {{{2^2 + (-1)^2 - 4*2 + 2*(-1) = -5}}}