I was trying to find the min/max for f(x,y) = x^3 + y^3 Subject to g(x,y) = xy = 4 .
Using Lagrange multipliers & partial derivatives. < 3x^2 , 3y^2 > = k< y ,x>
Solving these two + xy = 4, gives k = 6 , x = +-2 , y = +-2 This gives a max of 16 & min of -16 for f(x,y).
However these are obviously not the max, min. Basically the max, min are unbounded. So what am I missing here ? When is there a finite max, min for f(x,y) constrained by g(x,y) = c
A continuous image of a compact set is compact. A subset of $\mathbb R^n$ is compact iff it is closed and bounded. Unfortunately, these facts will not help much here because we're interested in the hehaviour of a conttinuous function on an hyperbola, which is closed but not bounded. It is certainly possible that a coninuous function might attain a maximum or a minimum on an unbounded set but there is no guarantee that that will happen and it does not happen in this case.