How can we find absolute extrema of $f(x,y,z)$ given that $g(x,y,z)=0$ ? I know how to find constrained extrema using Lagrange Multiplier but they are local extrema. Is there any theorem or conditions such that absolute extrema exist, and if so, how to identify global extrema ?
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Look at the closure of $S = { (x, y, z) \in \mathbb{R} : g(x, y, z) = 0 }$. If $S$ is compact you know there's a maximum and a minimum on it, and they are local extrema or they are on the edge of $S$. If it's not compact, you have for both the maximum and the minimum the possibilities: 1. It doesn't exist (i.e., there is a point in the closure such that f goes to +/- infinity approaching that point). 2. It does exist and is either on the border of $S$, or equal to a local extrema 3. It has no maximum or minimum, but it does have a supremum or infimum (consider for example the case 1/x > 0). – Ruben Dec 28 '15 at 09:25
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The tricky part, for non-compact domains, is excluding the case that there is an infimum or supremum (with value higher, or lower than each local extrema) that is not attained somewhere, so it is not a local extrema. There are some theorems to handle this, but not for the general case I think. Example: nonconstant holomorphic functions on $\mathbb{C}$ and nonconstant odd polynomials on $\mathbb{R}$, we know that they are unbounded ('in two directions', to $-\infty$ and $+\infty$) on their domains. – Ruben Dec 28 '15 at 09:39